Processing method and apparatus for processing spin echo in-phase and quadrature amplitudes from a pulsed nuclear magnetism tool and producing new output data to be recorded on an output record

ABSTRACT

A nuclear magnetic resonance (NMR) logging system includes a NMR logging tool adapted to be disposed in a borehole and a processing system adapted to be disposed on the surface of the borehole, the logging tool including a microcomputer, the processing system including a computer; and a signal processing system in accordance with the present invention having a first part stored in the downhole microcomputer of the logging tool and a second part stored in the surface oriented computer of the processing system. In the downhole microcomputer, using the first part of the signal processing system, a plurality of signal plus noise amplitudes are produced from a set of measured spin echo data, and a plurality N w  of the window sums I m ,m+1 are produced from the signal plus noise ampltudes, the window sums being transmitted uphole to the processing system. By producing the window sums, the first part of the signal processing system compresses and eliminates redundant information from the measured spin echo data. In the surface oriented processing system, using the second part of the signal processing system, the plurality N w  of the window sums I m ,m+1 are used to generate new data which is ultimately recorded on a new output record medium. Spectral amplitudes {a l  }, determined from the window sums, are used by the processing system to compute three log outputs and signal distributions presented as color maps. Three standard deviations are also computed. Log quality curves including signal phase and RMS noise are also computed.

BACKGROUND OF THE INVENTION

The subject matter of the present invention relates to a new method andapparatus for processing a signal which is output from a pulsed nuclearmagnetism tool disposed in a wellbore thereby producing new output datarepresentative of the formation traversed by the wellbore and forrecording the new output data on an output record medium.

Repeated attempts have been made to use the principles of nuclearmagnetic resonance by means of logging tools lowered through wellboresin oil exploration over the past several decades. It is recognized thatparticles of a formation having magnetic spin, for example atomicnuclei, protons, or electrons, have tendencies to align with a magneticfield which is imposed on the formation. Such a magnetic field may benaturally generated, as is the case with the earth's magnetic field(B_(E)) which has an intensity of approximately 0.5 gauss in areas ofthe globe where boreholes are typically drilled. Any given particle in aformation is additionally influenced by localized magnetic fieldsassociated with nearby magnetic particles, other paramagnetic materials,and the layer of ions which typically line pore walls of certain typesof formations such as shales. These localized fields tend to beinhomogeneous, while the earth's magnetic field is relativelyhomogeneous.

A nuclear magnetic resonance (NMR) logging tool apparatus, adapted to bedisposed in a borehole, produces a static and a substantiallyhomogeneous magnetic field focussed into a formation on one side of thelogging tool. By directing and configuring the combined magnetic fieldsof a plurality of magnets, a region, remote from the plurality ofmagnets, is introduced wherein the special field gradient substantiallyvanishes, thereby insuring that the field is highly homogeneousthroughout that region. In a preferred form, the magnets are mountedwithin a metallic skid or logging pad, the static magnetic field isdirected through the face of the pad into an adjacent formation, and theregion of substantially homogeneous field is situated in a volume offormation behind the mudcake layer which typically lines borehole walls.A homogeneous magnetic field, several hundred times stronger than theearth's magnetic field, can be thus imposed, or "focused", on a volumeof formation in situ.

Reference may be had to U.S. Pat. No. 4,933,638 issued Jun. 12, 1990 toKenyon et al (hereinafter termed, the "Kenyon et al patent" or "Kenyonet al") for details of such a nuclear magnetic resonance (NMR) loggingtool apparatus, which patent is incorporated herein by reference. In theKenyon et al patent, an RF antenna is mounted on the outside of thestructure of the pad so that the pad serves as a natural shield againstany signals which may be generated by resonant conditions behind thepad, particularly those potentially strong resonance signals fromborehole fluid. In the preferred form, the antenna is configured tofocus its signals radially outwardly from the pad face, into the volumeof formation having the homogeneous field, thereby additionally reducingthe distortion of measured signals from borehole effects.

All such nuclear magnetic resonance logging tool apparatus, whendisposed in a borehole, are electrically connected to a computingapparatus disposed at the surface of the borehole. The computingapparatus stores a signal processing software therein, the software inconjunction with the hardware of the computing apparatus producing aplurality output data representative of the characteristics of theformation traversed by the borehole when the software is executed by thehardware while utilizing a set of input data which was developed by thelogging tool disposed in the borehole.

While the prior art nuclear magnetic resonance logging tool of Kenyon etal, and its associated signal processing software, is capable ofdetermining formation characteristics with sufficient accuracy anddependability, it has been found useful to improve the performance andaccuracy of such logging tool, especially in view of the inherentdifficulties of making nuclear magnetic resonance (NMR) measurements inboreholes.

One very important improvement in the performance of the Kenyon et allogging tool can be made to the signal processing software used byKenyon et al. Several approaches to spectral decomposition or signalprocessing of NMR data have been reported. Spectral decomposition is asignal processing method that determines from NMR spin-echo signals inrocks, the individual amplitude components of the multi-exponentialsignal. These individual components correspond to different pore sizesin the rock.

A first approach is reported by Kenyon, W. E., Howard, J. J., Sezginer,A. Straley, C., Matteson, A., Horkowitz, R., and Erlich, R., in anarticle entitled "Poresize Distribution and NMR in Microporous ChertySandstones", Trans of the SPWLA of the 30th Ann. Logging Symp., PaperLL, Jun. 11-14, 1989, this first approach being further set forth in anarticle entitled "A NMR Technique for the Analysis of Pore Structure:Determination of Continuous Pore Size Distributions"; Gallegos, D. P.and Smith, D. M.; J. of Colloid and Interface Science, V. 122, No. 1,pp. 143-153, Mar. 1988, the disclosures of which are incorporated byreference into this specification.

A second approach is set forth in an article entitled "Problems inIdentifying Multimodal Distributions of Relaxation Times for NMR inPorous Media", Magnetic Resonance Imaging, Vol 9, pp. 687-693 (1991),the disclosure of which is incorporated by reference into thisspecification.

The aforementioned approaches are computationally too intensive to bedone in real time by a logging truck computer. They also do not compressthe data which is needed to limit the telemetry requirements for sendingdata uphole. In addition, the NMR logging tool should first acquiredownhole spin-echo measurements in earth formations penetrated by aborehole and then, secondly, generate a set of detailed formationevaluation information. However, this type of detailed formationevaluation information was previously obtainable only from costlylaboratory analysis of conventional core data. Therefore, a signalprocessing method and apparatus is needed which is adapted to extractthis formation evaluation information from the measured spin-echos. Thissignal processing method and apparatus must be capable of providing aspectral decomposition of the measured data, and it must be efficientand robust for real time processing of measured data during theacquisition of the data by an NMR logging tool moving in the borehole.Moreover, it is desirable to compress the data by elimination ofredundant information. This reduces the telemetry requirements of theNMR tool, which is important if the tool is run in combination withother logging tools.

SUMMARY OF THE INVENTION

Accordingly, it is a primary object of the present invention to providea new signal processing method and apparatus which is capable ofextracting a new set of specific formation evaluation information frommeasured spin-echo data acquired from a nuclear magnetic resonance (NMR)logging tool which is disposed in a borehole and moves within suchborehole.

It is a further object of the present invention to provide a new signalprocessing method and apparatus responsive to measured spin echo datafor providing a spectral decomposition of the measured spin-echo dataand subsequently generating a new set of specific formation evaluationinformation from the spectral decomposition, the signal processingmethod being efficient, flexible and robust for real time processing ofthe measured spin echo data during the acquisition of such data by a NMRlogging tool moving axially within the borehole.

It is a further object of the present invention to provide a new signalprocessing method and apparatus responsive to a set of measured spinecho data for extracting specific formation evaluation information fromthe measured spin echo data acquired from a NMR logging disposed in aborehole, the signal processing method and apparatus including a firstpart stored downhole in a downhole microcomputer for compressing andeliminating redundant information from the set of measured spin echodata and generating a plurality of window sums, where the number of theplurality of window sums is much less than the number of the set ofmeasured spin echo data, the plurality of window sums being transmitteduphole, and a second part stored uphole in a surface oriented computersystem and responsive to the plurality of window sums received from thedownhole microcomputer for generating a new set of specific formationevaluation information representative of characteristics of a formationtraversed by the borehole.

It is a further object of the present invention to provide the newsignal processing method and apparatus which generates the new set ofspecific formation evaluation information representative ofcharacteristics of the formation traversed by the borehole, which newset of specific formation evaluation information includes total NMRporosity, free fluid porosity, bound fluid porosity, spin-spin (T2)relaxation time distributions which are related to pore sizedistributions in the formation, and continuous permeability logs.

It is a further object of the present invention to provide the newsignal processing method and apparatus which generates the new set ofspecific formation evaluation information, which specific formationevaluation information includes the following information: total NMRporosity, free fluid porosity, porosity standard deviation, free fluidstandard deviation, and measurement diagnostics including RMS noiseestimate and signal phase.

It is a further object of the present invention to provide the newsignal processing method and apparatus which provides the aforementionednew set of specific formation evaluation information and which generatesa new output record medium illustrating the specific formationevaluation information, the new output record medium comprising aplurality of logs, each log illustrating a particular one of theaforementioned set of specific formation evaluation information andmeasurement diagnostics as a function of depth in the borehole.

In accordance with these and other objects of the present invention, anuclear magnetic resonance (NMR) logging system includes: (1) a NMRlogging tool adapted to be disposed in a borehole, the logging toolincluding a downhole microcomputer; (2) a processing system adapted tobe disposed at the surface of the borehole and electrically connected tothe logging tool for processing signals received from the logging tool,the processing system including a computer having a memory; and (3) asignal processing method and apparatus embodied in the form of asoftware package having a first part stored downhole in the downholemicrocomputer of the logging tool for compressing and eliminatingredundant information from received and measured spin echo data andgenerating compressed information; and a second part stored uphole inthe surface oriented computer of the processing system and responsive tothe compressed information for generating the new set of specificformation evaluation information and for further generating an outputrecord medium which illustrates the specific formation evaluationinformation.

In the logging tool disposed in the borehole, a receiving antennameasures voltages induced by the precession of the magnetic moments ofindividual protons in the volumes of the formation traversed by theborehole and generates a plurality of spin-echo receiver voltage pulsesrepresentative of the magnetic moments. An electronics cartridge, whichincludes a microcomputer, begins processing the receiver voltage pulsesby integrating each of the spin-echo receiver voltage pulses over a timeinterval, there being a total of J time intervals, each time intervalbeing centered about a time t_(j) =j delta, where j=1, 2, . . . , J. Theintegrated signals are recorded as spin-echo inphase (R_(j)) andquadrature (X_(j)) amplitudes, time series channels or waveforms. In theDESCRIPTION OF THE PREFERRED EMBODIMENT, the symbols "R_(j) " and "X_(j)" will be written without a "tilde symbol overbar", whereas the samesymbols in the DETAILED DESCRIPTION will be written with the tildesymbol overbar.

In the microcomputer, using the first part of the signal processingmethod and apparatus in accordance with the present invention, a signalphase (theta) is estimated from the 2J spin-echo in-phase (R_(j)) andquadrature (X_(j)) amplitudes associated with the 2J spin-echo receivervoltage pulses. Then, the in-phase (R_(j)) amplitude, quadrature (X_(j))amplitude, and the signal phase (theta) associated with each of thespin-echo receiver voltage pulses are combined to produce a signal plusnoise amplitude A_(j).sup.(+). A plurality of the signal plus noiseamplitudes A_(j).sup.(+) are disposed within a time window, there beinga plurality of time windows. A sum of the plurality of signal plus noiseamplitudes A_(j).sup.(+) within each time window produces a window sumI_(m),m+1 ; and, since there are a plurality of time windows, there area plurality of window sums I_(m),m+1. Consequently, the downholemicrocomputer of the NMR logging tool disposed in the borehole generatesa plurality of the window sums I_(m),m+1, one for each time window, andtransmits the plurality of the window sums uphole to the processingsystem located at the surface of the borehole. Note that each window sumI_(m),m+1 itself represents a reduced set of data, since each window sumI_(m),m+1 is the sum of a plurality of signal plus noise amplitudesA_(j).sup.(+). As a result, the telemetry requirements needed by thelogging tool to transmit the plurality of window sums I_(m),m+1 upholeto the processing system located at the wellbore surface aresubstantially reduced. In addition to producing the signal plus noiseamplitude A_(j).sup.(+), the downhole microcomputer also produces a setof "J" amplitudes A_(j).sup.(-). These amplitudes A_(j).sup.(-) are usedby the downhole microcomputer for estimating an RMS noise, which isdefined to be the square root of "psi", where "psi" is the noise power.The RMS noise is also transmitted uphole to the processing systemdisposed at the wellbore surface.

In the processing system disposed at the wellbore surface, using thesecond part of the signal processing method and apparatus in accordancewith the present invention, the RMS noise is used to compute threestandard deviations: the standard deviation "sigma(phi_(nmr))", thestandard deviation "sigma(phi_(f))", and the standard deviation"sigma(T₂,log)". These standard deviations are used to generate the newoutput record medium in accordance with the present invention. The RMSnoise is also used to compute the dimensionless parameter "gamma", theuse of which is discussed below. As noted earlier, the plurality N_(w)of the window sums I_(m),m+1 are transmitted uphole by the downholecomputer. The processing system located at the wellbore surface includesa surface computer. The surface computer receives the plurality N_(w) ofthe window sums I_(m),m+1 ; and, using the plurality N_(w) of windowsums I_(m),m+1 and the aforementioned dimensionless parameter "gamma",the surface computer determines a logarithm of the likelihood function(-1n L) for the N_(w) window sums set forth in equation 42 in thedetailed description of the preferred embodiment. The logarithm of thelikelihood function (-1n L) of equation 42 is a function of a set ofspectral amplitudes "a₁ ", where such spectral amplitudes are determinedby minimization of equation 42. The spectral amplitudes "a₁ " are usedby the surface computer: (1) to compute three log outputs, phi_(nmr),phi_(f), and T₂,log, and (2) to compute signal distributions P_(a)(logT₂) which are represented by color maps.

The new output record medium in accordance with the present invention isgenerated using the three log outputs, the signal distributions forcolor maps, and the three standard deviations.

Further scope of applicability of the present invention will becomeapparent from the detailed description presented hereinafter. It shouldbe understood, however, that the detailed description and the specificexamples, while representing a preferred embodiment of the presentinvention, are given by way of illustration only, since various changesand modifications within the spirit and scope of the invention willbecome obvious to one skilled in the art from a reading of the followingdetailed description.

BRIEF DESCRIPTION OF THE DRAWINGS

A full understanding of the present invention will be obtained from thedetailed description of the preferred embodiment presented hereinbelow,and the accompanying drawings, which are given by way of illustrationonly and are not intended to be limitative of the present invention, andwherein:

FIG. 1 illustrates a nuclear magnetic resonance (NMR) logging systemincluding a NMR logging tool disposed in a wellbore and a processingsystem disposed at the wellbore surface for processing signalstransmitted uphole by the logging tool;

FIG. 2 illustrates a sketch of the logging tool in the wellboreproducing a static magnetic field B0 into a formation traversed by thewellbore;

FIGS. 3-5 illustrate sections of the formation adjacent the logging toolin the wellbore and the effects of the magnetic field B0 on thecomposite magnetic moment vector M representative of the sum of aplurality of individual magnetic moments associated with a correspondingplurality of protons disposed within elements of the formation;

FIGS. 6-7 illustrate the effects on the proton's composite magneticmoment M of the presence of two magnetic fields: the static magneticfield B0 and a radio frequency magnetic field B_(x90), the direction ofwhich is transverse to the static magnetic field B0 and along thex-axis, the field B_(x90) being applied along the x-axis for a duration"T90" which causes a 90 degree rotation of the composite magnetic momentvector M about the x-axis;

FIGS. 7a-7b illustrate the precession of the individual magnetic momentsvector m1 and the effects on this magnetic moments vector m1 of thepresence of two other magnetic fields: the static magnetic field B0 anda radio frequency magnetic field By₁₈₀, the direction of which istransverse to the magnetic field B0 and is directed along the y-axis;

FIGS. 8a-8m illustrate the effects on the precession rate of anindividual magnetic moment "A_(n) " which is located within a strongerfield strength of the static magnetic field B0, relative to anotherindividual magnetic moment "A_(f) " which is located within a weakerfield strength of the static magnetic field B0, since the field B0 isnon-homogeneous and the individual magnetic moments precess at differentrates about the z-axis, the direction of the magnetic field B0 vector,and the effects on the magnetic moments A_(n) and A_(f) of a furthermagnetic field pulse B_(y180) which "rotates" the magnetic moments A_(n)and A_(f) 180 degrees about the y-axis thereby causing the magneticmoments A_(n) and A_(f) to refocus and to produce the spin-echo receivervoltage pulses of FIG. 9b;

FIGS. 9a-9b illustrate the spin-echo transmitter and receiver voltagepulses produced by the logging tool in time relation to: (1) a firstmagnetic field pulse which is directed along the x-axis of FIG. 6 andhas a pulse duration of T90 (the first magnetic field pulse beinghereinafter termed "B_(x90) " or "the 90 pulse"), and (2) a secondmagnetic field pulse directed along the y-axis and having a pulseduration of T180 which rotates all of the proton spins 180 degrees aboutthe y-axis (the second magnetic field pulse being hereinafter termedB_(y180));

FIG. 10 illustrates NMR logging tool disposed in a borehole including anelectronics cartridge which stores the first part of the signalprocessing method and apparatus in accordance with the presentinvention, and the processing system disposed on the surface of theborehole which stores the second part of the signal processing methodand apparatus in accordance with the present invention;

FIG. 11 illustrates a more detailed construction of the surface orientedprocessing system of FIG. 10 which stores the second part of the signalprocessing method and apparatus of the present invention;

FIGS. 12a, 12b1 and 12b2 illustrate a more detailed construction of theelectronics cartridge of FIG. 10 which stores the first part of thesignal processing method and apparatus of the present invention;

FIG. 13 illustrates a block diagram or flow chart of the first part ofthe signal processing method and apparatus of the present inventionstored in the electronics cartridge of the NMR tool and the second partof the signal processing method and apparatus of the present inventionstored in the surface oriented processing system of FIG. 10, which firstpart and second part of the signal processing method and apparatus isembodied in the form of software stored,, in a memory of the electronicscartridge and the surface oriented processing system, respectively;

FIG. 14 illustrates a plurality of the spin-echo signal plus noiseamplitudes A_(j).sup.(+) corresponding, respectively, to the pluralityof spin-echo signals of FIG. 9b, each spin-echo signal plus noiseamplitude A_(j).sup.(+) being identified by an echo number, the echonumbers being divided into a plurality of time windows;

FIG. 15 illustrates a plurality of window sums I_(m),m+1 corresponding,respectively, to the plurality of time windows of FIG. 14;

FIG. 16 illustrates a new output record medium, comprising a pluralityof new logs, generated by the processing system disposed at the,wellbore surface in accordance with another aspect of the presentinvention;

FIG. 16a illustrates a more detailed block diagram of the surfacecomputer and the downhole computer of FIG. 12a utilized in conjunctionwith FIG. 13 for providing a functional description of the presentinvention;

FIG. 17 illustrates the standard deviation in porosity estimates versusthe total number of echos for two values of "gamma";

FIG. 18 illustrates the standard deviation in porosity estimates versusthe total number of echos for two values of t_(cp) ;

FIG. 19 illustrates the standard deviation in porosity estimates versusthe total number of echos for two values of W:

FIG. 20 illustrates the standard deviation in porosity estimates versusthe total number of echos for three values of T_(min) ;

FIG. 21 illustrates a signal distribution computed from station data at1100 ft for two values of gamma differing by an order of magnitude;

FIG. 22 illustrates a signal distribution at 1125 ft exhibiting a twopeak structure reflecting two disparate pore size distributions;

FIG. 23 illustrates a signal distribution at 1148 ft indicating arelatively poor reservoir quality rock;

FIG. 24 illustrates a signal distribution at 1200 ft indicating arelatively good quality permeable reservoir rock;

FIG. 25 illustrates a signal distribution at 1229 ft indicatingessentially no bound fluid, and a long relaxation time; and

FIG. 26 illustrates a correction "delta phi" used in equation C.1associated with a calculation of "delta phi".

DESCRIPTION OF THE PREFERRED EMBODIMENT

This specification is divided into two parts:

(1) a Description of the Preferred Embodiment, which provides a general,more understandable description of the new signal processing method andapparatus of the present invention; and

(2) a Detailed Description of the Preferred Embodiment, which provides amore detailed description of the new signal processing method andapparatus of the present invention.

Referring to FIG. 1, a nuclear magnetic resonance (NMR) logging systemis illustrated, the NMR logging system including a NMR logging tool 13disposed in a wellbore and a processing system 7a disposed at thewellbore surface for processing signals transmitted uphole by thelogging tool.

In FIG. 1, a borehole 10 is shown adjacent to formations 11, 12, thecharacteristics of which are to be determined. Within borehole 10 thereis shown a logging tool 13 connected via a wireline 8 to surfaceequipment 7. The surface equipment 7 includes a processing system 7awhich stores therein a signal processing method and apparatus embodiedin the form of software. The processing system 7a will be discussed inmore detail with reference to FIGS. 12 and 13 of the drawings. Tool 13preferably has a face 14 shaped to intimately contact the borehole wall,with minimal gaps or standoff. The tool 13 also has a retractable arm 15which can be activated to press the body of the tool 13 against theborehole wall during a logging run, with the face 14 pressed against thewall's surface. Although tool 13 is shown in the preferred embodiment ofFIG. 1 as a single body, the tool may obviously comprise separatecomponents such as a cartridge, sonde or skid, and the tool may becombinable with other logging tools as would be obvious to those skilledin the art. Similarly, although wireline 8 is the preferred form ofphysical support and communicating link for the invention, alternativesare clearly possible, and the invention can be incorporated in a drillstem, for example, using forms of telemetry which may not require awireline. The formations 11, 12 have distinct characteristics such asformation type, porosity, permeability and oil content, which can bedetermined from measurements taken by the tool. Deposited upon theborehole wall of formations 11, 12 is typically a layer of mudcake 16which is deposited thereon by the natural infiltration of borehole fluidfiltrate into the formations. In the preferred embodiment shown in FIG.1, tool 13 comprises a first set magnet array 17 and an antenna 18positioned between the array 17 and the wall engaging face 14. Magnetarray 17 produces a static magnetic field B0 in all regions surroundingthe tool 13. The antenna 18 produces, at selected times, an oscillatingradio frequency magnetic field Bl (previously denoted as "B_(x90) " andB_(y180) ") which is focussed into formation 12, and is superposed onthe static field B0 within those parts of formation opposite the face14. The field Bl is perpendicular to the field B0. The Volume ofInvestigation, 19, of the tool for the first set magnet array 17 shownin dotted lines FIG. 1, is a vertically elongated region directly infront of tool face 14 in which the magnetic field produced by the magnetarray 17 is substantially homogeneous and the spatial gradient thereofis approximately zero. The tool 13 may also comprise, as an option, asecond set magnet array 20 and an antenna 21 positioned between thearray 20 and the wall engaging face 14. Magnet array 20 produces anotherstatic magnetic field B0 in all regions surrounding the tool 13. Theantenna 21 produces, at selected times, an oscillating radio frequencymagnetic field Bl which is again focussed into formation 12, and issuperposed on the static field B0 within those parts of formationopposite the face 14. The Volume of Investigation 22 of the tool for thesecond set magnet array 20, shown in dotted lines in FIG. 1, is avertically elongated region directly in front of tool face 14 in whichthe magnetic field produced by the magnet array 20 is substantiallyhomogeneous and the spatial gradient thereof is approximately zero. Dueto the particular magnet arrangement for the second set magnet array 20,the Volume of Investigation 22 is at a depth in the formation 12 whichis greater than the depth at which the Volume of Investigation 19 islocated. A prepolarizing magnet 23, shown in dotted lines, may bepositioned directly above the array 17 in a modified embodiment of theinvention in accordance with the teachings of the aforementioned Kenyon,et al patent. An electronics cartridge 24 is positioned above the magnet23. The electronics cartridge 24 includes a downhole microcomputer. Theelectronics cartridge 24, including the downhole microcomputer, will bediscussed in more detail with reference to FIGS. 12a-12b of thedrawings.

In operation, referring to FIG. 1, the tool 13 makes a measurement inthe Volume of Investigation 19 by magnetically reorienting the nuclearspins of particles in formation 12 with a pulse of oscillating magneticfield Bl (previously denoted as "B_(x90) " and B_(y180) "), and thendetecting the precession of the tipped particles in the static,homogeneous field BO within the Volume of Investigation 19, over aperiod of time. As seen in FIG. 1, this Volume of Investigation does notoverlap the surface of the wall engaging face 14 as in some previouslogging tools, and does not overlap the mudcake 16 on the borehole wall.In a pulse echo type of measurement, a pulse of RF current is passedthrough the antenna 18 to generate a pulse of RF field Bl where the RFfrequency is selected to resonate only hydrogen nuclei subjected to astatic magnetic field strength equal or nearly equal to the field B0within the Volume of Investigation 19. The signals induced in antenna 18subsequent to the RF pulse Bl represent a measurement of nuclearmagnetic precession and decay within the Volume, automatically excludingany undesirable contributions from the borehole fluid, mudcake, orsurrounding formations where the field strength of B0 is substantiallydifferent. The tool 13 makes a measurement in the Volume ofInvestigation 22 in the same manner discussed above with respect to theVolume of Investigation 19 but utilizing the second set magnet array 20and the antenna 21.

The general principles associated with nuclear magnetic resonance welllogging, utilized by the NMR logging system of FIG. 1, will be discussedin the following paragraphs with reference to FIGS. 2 through 10 of thedrawings.

In FIG. 2, a NMR tool 13 contacts a wall of the borehole 10. A magnetdisposed within the tool 13 generates a static magnetic field B0, whichfield B0 is directed toward a volume of investigation 19 disposed withina portion of the formation 12 traversed by the borehole 10.

FIG. 3 illustrates a cross section 19a of the volume of investigation 19of FIG. 2. The cross section 19a includes a plurality of pores 19al,each of which contain oil and/or water, the oil and/or water beingfurther comprised of a plurality of protons, each proton having amagnetic spin or magnetic moment (u_(i)) identified in FIG. 3 by thearrow A shown in FIG. 3. Since there are a plurality of protons in eachof the pores 19al, there are a corresponding plurality of magneticmoments (u₁, u₂, u₃, . . . , u_(n)) in each pore 19al, where "n" equalsthe number of protons in each pore.

FIG. 4 illustrates a plurality of magnetic moments "u_(i) " (identifiedin FIG. 4 by a plurality of arrows "A" ) associated with a correspondingplurality of protons disposed within a particular one of the pores 19alof the volume of investigation 19 when the static magnetic field B0 iszero. Note that the magnetic moments (u_(i)) A all point in differentdirections. Therefore, a composite magnetic moment "M", defined to bethe vector sum of all individual magnetic moments A (that is, M=u₁ +u₂+u₃ +. . . +u_(n)) is approximately zero.

However, FIG. 5 illustrates the same plurality of magnetic moments A ofFIG. 4, but now the magnetic field B0 is not equal to zero. Note that,in FIG. 5, the magnetic moments (u_(i)) A align together in the samedirection when the magnetic field B0 is not equal to zero. Therefore,the composite magnetic moment "M" is not approximately equal to zero,but rather, is equal to a sum which is defined to be the sum of allindividual magnetic moments A. Since each individual magnetic moment Acan be quantified by the term "u_(i) ", the composite magnetic moment"M" is equal to the sum of all individual magnetic moments "u_(i) ",where i=1,2,3 . . . ,n, associated with the "n" individual protonsdisposed within a pore space 19al associated with the volume ofinvestigation 19, that is, M=u₁ +u₂ +u₃ +. . . +u_(n).

Referring to FIG. 6, assume that the composite magnetic moment "M" forthe volume of investigation 19 of FIG. 3-5 is aligned along the "z"axis; assume further that the static magnetic field B0 is also alignedalong the z-axis. Then, assume that an oscillating radio frequencymagnetic field pulse "B1" or "B_(x90) " is applied along the x-axis,transverse to the z-axis.

Referring to FIG. 7, recalling the assumptions mentioned above withreference to FIG. 6, since the pulse duration of the oscillatingmagnetic field pulse "B_(x90) " is 90 degrees and is applied along thex-axis in the presence of the static magnetic field B0, the compositemagnetic moment "M" rotates 90 degrees from the z-axis to the y-axis, asshown in FIG. 7.

Referring to FIGS. 7a-7b, the magnetic moment vector M is initiallydisposed along the y-axis as shown in FIG. 7, and begins to "precess"(or rotate) clockwise within the x-y plane, as shown in FIG. 7a.However, in FIG. 7b, if an oscillating magnetic field pulse B_(y180) isapplied along the y-axis (for a period of time equivalent to a 180degree pulse duration), the magnetic moment vector M rotates or flips180 degrees from one quadrant within the x-y plane to another quadrantwithin the x-y plane, as illustrated by element numeral 30 in FIG. 7b.The significance of this concept will become clear with reference toFIGS. 8a-8l of the drawings.

Referring to FIG. 8a-8m, recall that the composite magnetic moment "M"is defined to be the vector sum of all the individual magnetic moments"u_(i) " (identified by the letter A) of all protons within a pore space19al in FIG. 3, that is, M=SUMMATION u_(i), where i=1, 2, . . . , n, orM=u₁ +u₂ +u₃ +. . . +u_(n). In FIGS. 7a-7b, it was shown that, in thepresence of the static magnetic field B0, the composite magnetic momentvector M precessed in the x-y plane, then flipped to another quadrant inresponse to the B_(y180) oscillating magnetic field pulse.

Since the composite magnetic moment vector M is the sum of allindividual magnetic moments "u_(i) ", in actuality, all of theindividual magnetic moments "u_(i) ", being disposed within the staticmagnetic field B0, individually precess in the x-y plane; and, then,each individual magnetic moment "u_(i) " flips to another quadrant inresponse to the B_(y180) oscillating magnetic field pulse.

However, the static magnetic field B0 is not homogeneous, that is, someparts of the static magnetic field B0 are stronger in terms of fieldstrength than other parts of the field B0. Therefore, if one individualmagnetic moment "u₁ " is disposed within a stronger part of the staticmagnetic field B0, and another individual magnetic moment "u₂ " isdisposed within a weaker part of the static magnetic field B0, the rateof precession or rotation within the x-y plane of the one magneticmoment "u₁ " will be greater than the rate of precession or rotationwithin the x-y plane of the other magnetic moment "u₂ ".

FIGS. 8a-8m illustrate the effects on the precession rate of the oneindividual magnetic moment "u₁ " which is located within a stronger partof the field B0 relative to another individual magnetic moment "u₂ "which is located within a weaker part of the field B0; furthermore,FIGS. 8a-8l illustrate the effects on the one individual magnetic moment"u₁ " and the other individual magnetic moment "u₂ " of a further,oscillating magnetic field pulse B_(y180) which "rotates" or "flips" themagnetic moments "u₁ " and "u₂ " 180 degrees about the y-axis therebycausing the magnetic moments "u₁ " and "u₂ " to refocus and to produceone of the spin-echo receiver voltage pulses of FIG. 9b.

In FIG. 8a, the plurality of individual magnetic moments "u_(i) " A (andthus, the composite magnetic moment M) rotate 90 degrees in the z-yplane, as shown in FIGS. 6-7 of the drawings, in response to the static,radio frequency magnetic field pulse "B_(x90) " which is applied alongthe x-axis for a time equivalent to a 90 degree pulse duration.

In FIG. 8b, the one individual magnetic moment "u₁ ", the magneticmoment associated within one particular proton which is disposed withina stronger part of the magnetic field B0, precesses or rotates, in aclockwise direction, at a faster rate than does the other individualmagnetic moment "u₂ ", the other individual magnetic moment associatedwith another proton that is located within a weaker part of the magneticfield B0.

In FIGS. 8c and 8m, a further, oscillating magnetic field pulse B_(y180)is applied at a time t_(cp) after application of the magnetic fieldpulse B_(x90) (which is assumed to be applied at a time t=0) along they-axis for a period of time equivalent to a pulse duration of 180degrees; in response to the further magnetic field pulse B_(y180), theindividual magnetic moments "u₁ " and "u₂ " rotate or flip about they-axis by an amount equal to 180 degrees thereby removing the twoindividual magnetic moments "u₁ " and "u₂ " from quadrant number 2 andlocating the two individual magnetic moments "u₁ " and "u₂ " in quadrantnumber 1 of the x-y plane, as shown in FIG. 8c.

In FIG. 8d, both of the individual magnetic moments "u₁ " and "u₂ "continue to rotate in the clockwise direction but the one magneticmoment "u₁ " continues to rotate or precess at a faster rate than doesthe other magnetic moment "u₂ ".

In FIG. 8e, since, as shown in FIG. 8c, the two individual magneticmoments "u₁ " and "u₂ " flipped from quadrant 2 to quadrant 1 of the x-yplane in response to the oscillating magnetic field B_(y180), the twoindividual magnetic moments "u₁ " and "u₂ " refocus (align together asone magnetic moment vector) thereby producing a first spin-echo signal(echo 1) at a time 2t_(cp), which spin-echo signal is picked up by theantennas 18 or 21 of the NMR tool 13 of FIG. 1 and is transmitted upholeto the processing system 7a. The first spin-echo signal (echo 1) isshown in FIG. 9b of the drawings.

In FIG. 8f, the one magnetic moment "u₁ " proceeds to precess ahead orin front of the other magnetic moment "u₂ " since the rate of precessionof the one magnetic moment "u₁ " is still greater than the rate ofprecession of the other individual magnetic moment "u₂ ".

In FIGS. 8g and 8m, a further oscillating magnetic field pulse B_(y180)is applied at a time 3t_(cp) along the y-axis for a period of timeequivalent to a pulse duration of 180 degrees; in response to thefurther magnetic field pulse B_(y180), the two individual magneticmagnetic moments "u₁ " and "u₂ " rotate or flip about the y-axis by anamount equal to 180 degrees (similar to the action depicted in FIG. 8c)thereby removing the two magnetic moments "u₁ " and "u₂ " from quadrantnumber 2 and locating the two magnetic moments "u₁ " and "u₂ " intoquadrant number 1 of the x-y plane, as shown in FIG. 8g.

In FIG. 8h, the two individual magnetic moments "u₁ " and "u₂ " refocusagain, similar to the action depicted in FIG. 8e, thereby producing asecond spin-echo signal (echo 2) at a time 4t_(cp), which secondspin-echo signal is picked up by antennas 18 and 21 of the NMR tool 13of FIG. 1 and transmitted uphole to the processing system 7a. FIG. 9billustrates the echo 2, second spin-echo signal. In FIG. 8i, the oneindividual magnetic moment "u₁ " proceeds ahead of the other individualmagnetic moment "u₂ " since the rate of precession of the one magneticmoment "u₁ " is greater than the rate of precession of the othermagnetic moment "u₂ ". In FIGS. 8j and 8m, the two individual magneticmoments "u₁ " and "u₂ " flip or rotate 180 degrees about the y-axis fromquadrant 2 to quadrant 1 of the x-y plane in response to application ofthe oscillating magnetic field pulse B_(y180), applied along the y-axisat a time 5t_(cp).

In FIG. 8k, the two individual magnetic moments "u₁ " and "u₂ " refocusagain thereby producing a third spin-echo signal (echo 3) at a time6t_(cp), which third spin-echo signal (echo 3) is picked up by antennas18 and 21 of NMR tool 13 and transmitted uphole to processing system 7a.The echo 3, third spin-echo signal is illustrated in FIG. 9b.

In FIG. 8l, the one individual magnetic moment "u₁ " proceeds ahead ofthe other individual magnetic moment "u₂ " since the rate of precessionof the one magnetic moment "u₁ " is greater than the rate of precessionof the other magnetic moment "u₂ ".

Therefore, three spin-echo signals have been produced, the firstspin-echo signal (echo 1) being produced in connection with FIG. 8e, thesecond spin-echo signal (echo 2) being produced in connection with FIG.8h, and the third spin-echo signal (echo 3) being produced in connectionwith FIG. 8k.

Referring to FIGS. 9a-9b, the aforementioned first and second spin-echosignals, echo 1 and echo 2, are illustrated in time relation to theoscillating magnetic field pu₁ se B_(x90) (90 pulse) which is directedalong the x-axis of FIG. 6 and has a pulse duration of 90 degrees, andto the further oscillating magnetic field pulse B_(y180) (180 pulse)directed along the y-axis and having a pulse duration of 180 degrees.FIG. 9b illustrates a plurality of spin-echo signals associated witheither the (R_(j)) inphase channel or the (X_(j)) quadrature channel.When FIGS. 9a-9b are examined jointly with FIGS. 8a-8m, it is evidentthat the nth spin-echo signal is produced at a time 2(n)t_(cp) followingapplication of the magnetic field pulses B_(y180) at (2n-1)t_(cp), wheren=1, 2, 3, . . . , J. The first spin-echo signal (echo 1) is generatedafter a time t_(cp) elapses following application of the first magneticfield pulse B_(y180) and the second spin-echo signal (echo 2) isgenerated after a time t_(cp) elapses following application of thesecond magnetic field pulse B_(y180).

Referring to FIG. 10, a nuclear magnetic resonance (NMR) logging systemis illustrated, the logging system including a NMR logging tool 13disposed in a wellbore and surface equipment 7 in the form of a welllogging truck 7 situated on the surface of the wellbore, the welllogging truck 7 including a processing system 7a in the form of acomputer 7a situated within the well logging truck 7.

In FIG. 10, the NMR logging tool 13 includes antennas 18 and 21 and anelectronics cartridge 24 responsive to signals received by the antennas18 and 21 for processing the signals before transmission of the signalsuphole to the computer 7a in the well logging truck. The electronicscartridge 24 of the logging tool 13 stores a first part of the signalprocessing (software) method and apparatus in accordance with one aspectof the present invention, and the computer 7a stores a second part ofthe signal processing (software) method and apparatus in accordance withanother aspect of the present invention.

In FIG. 11, the computer 7a includes a system bus 7a1, a processor 7a3connected to the system bus 7a1 for generating new output data inaccordance with one aspect of the present invention, a memory 7a4connected to the system bus 7a1 for storing the second part of thesignal processing (software) method and apparatus of the presentinvention, and a recorder 7a2 connected to the system bus for receivingthe new output data from the processor and generating a new outputrecord medium 7a2A, to be discussed with reference to FIG. 16 of thedrawings, in accordance with another aspect of the present invention.The computer 7a may include or consist of any one of the followingcomputer systems manufactured by Digital Equipment Corporation (DEC),Maynard, Mass.: (1) DEC VAX 6430, (2) DEC PDP-11, or (3) DEC Vaxstation3100, or it may include any other suitable computer system.

Referring to FIGS. 12a, 12b1, and 12b2, a contruction of the electronicscartridge 24 of FIG. 10 is illustrated. A more detailed construction ofthe hardware associated with the NMR logging system of FIGS. 1, 12a,12b1, and 12b2, and in particular, of the construction of theelectronics cartridge 24, is set forth in a prior pending applicationentitled "Borehole Measurement of NMR Characteristics of EarthFormations", corresponding to allowed application Ser. No. 07/970,324filed Nov. 2, 1992, the disclosure of which is incorporated by referenceinto this specification.

In FIG. 12a, the surface computer 7a is electrically connected vialogging cable to the electronics cartridge 24 of FIG. 10. Theelectronics cartridge 24 includes a downhole computer 46. The downholecomputer 46 is ultimately electrically connected to the antennas 18 and21.

In FIGS. 12b1, the downhole computer 46 of electronics cartridge 24includes a system bus 46b to which a processor 46c is electricallyconnected and a memory 46a is electrically connected.

In FIG. 12b2, the memory 46a stores the first part of the signalprocessing method and apparatus of the present invention.

Referring to FIG. 13, a flow chart of the first part and the second partof the signal processing (software) method and apparatus of the presentis illustrated.

This specification is divided into a Description of the PreferredEmbodiment, and a Detailed Description of the Preferred Embodiment. Theflow chart of FIG. 13 and the following discussion is part of theDescription of the Preferred Embodiment and provides a generaldiscussion of the signal processing method and apparatus of the presentinvention. The Detailed Description of the Preferred Embodiment setforth below provides a more detailed discussion of the aforementionedsignal processing method and apparatus of the present invention. In thefollowing general discussion, occasional reference to equations andother specific description set forth in the Detailed Description of thePreferred Embodiment will be required.

The signal processing method and apparatus of the present invention,embodied in the form of a software package, includes two parts: a firstpart 46a1 stored in the memory 46a of the downhole computer 46 ofelectronics cartridge 24 of FIGS. 12a-12b2 and a second part 7a4A storedin the memory 7a4 of the well logging truck computer 7a of FIG. 10situated at the uurface of the wellbore.

In FIG. 1, the receiving antennas 18 and 21 of logging tool 13 measurethe magnetic moments of individual protons in the volumes 19 and 22 ofthe formation traversed by the borehole 10 and generate a plurality ofspin-echo receiver voltage pulses (FIG. 9b) representative of themagnetic moments. The electronics cartridge 24 begins processing thetwo-channel receiver voltage pulses by integrating each of the spin-echoreceiver voltage pulses over a time interval, there being a total of Jtime intervals, each time interval being centered about a time t_(j) =jdelta, where j=1, 2, . . . ,J. The integrated signals are recorded asspin-echo inphase (R_(j)) and quadrature (X_(j)) amplitudes, time serieschannels or waveforms.

Referring to FIG. 13, the first part 46a1 of the signal processingmethod and apparatus of the present invention is illustrated.

In FIG. 13, the first part 46a1 of the signal processing method andapparatus of the present invention receives the aforementioned inphase(R_(j)) and quadrature (X_(j)) amplitudes, and a signal phase (theta) isestimated associated with these amplitudes, block a1A. Equation 9 of theDetailed Description set forth below provides the equation of the signalphase (theta) as a function of the inphase (R_(j)) and quadrature(X_(j)) amplitudes, as follows: ##EQU1## Since the inphase (R_(j))amplitude, the quadrature (X_(j)) amplitude, and the signal phase(theta) is known for each spin-echo receiver voltage pulse, the signalplus noise amplitude A_(j) (+) and the amplitude A_(j).sup.(-), for eachspin-echo receiver voltage pulse, may now be determined, block a1B, byutilizing equations 22 and 23 of the Detailed Description, as follows:

    A.sub.j.sup.(+) =R.sub.j cos θ+X.sub.j sin θ,  (22)

    A.sub.j.sup.(-) =R.sub.j sin θ-X.sub.j cos θ.  (23)

The RMS noise, the square root of psi or "SQRT psi", can be estimatedfrom the signal plus noise amplitude A_(j).sup.(-), block alC, byutilizing a practical implementation of equation 31 of the DetailedDescription, as follows:

    1J SUMMATION (j=1 . . . J) (A.sub.j.sup.(-)).sup.2 approximately =psi,

where equation (31) is set forth as follows:

    <(A.sub.j.sup.(-)).sup.2 >≃ψ,            (31)

The window sum I_(m),m+1 can be computed from the signal plus noiseamplitude A_(j).sup.(+), block a1D, by utilizing equations 22 and 35 ofthe Detailed Description, as follows: ##EQU2##

As a result, when the downhole computer 46 of FIG. 12 executes the firstpart 46a1 of the signal processing software of FIG. 13, two outputs aregenerated: the window sum I_(m),m+1 which is determined from equation 35and the RMS noise (SQRT psi) which is determined from equation 31.

The following paragraphs present a more detailed explanation of thewindow sum I_(m),m+1 determined from equation 35.

Referring to FIG. 14, an example of signal plus noise amplitudesA_(j).sup.(+) that have been determined from spin-echo signals (echo 1,echo 2,...) like those shown schematically as either R or X channelspin-echo pulses in FIG. 9b is illustrated in FIG. 14. The spin-echosignal plus noise amplitudes A_(j).sup.(+) are separated in time by"2t_(cp) " from each other.

Referring to FIGS. 14 and 15, the technique for determining a particularwindow sum (I_(m),m+1) from a plurality of signal plus noise amplitudes(A₁.sup.(+), A₂.sup.(+), A₃.sup.(+), A₄.sup.(+), . . . , A_(n).sup.(+))is illustrated.

Recalling that a window sum I_(m),m+1 is determined from the equation"I_(m),m+1 =SUMMATION A_(j).sup.(+) ", as set forth in equation 35 ofthe Detailed Description of the Preferred Embodiment set forth below,referring to FIGS. 14 and 15, a first window sum "I₁,2 ", where m=1, maybe determined by summing a plurality of individual signal plus noiseamplitudes A₁.sup.(+), A₂.sup.(+), A₃.sup.(+), A₄.sup.(+), . . . ,A_(n).sup.(+) which are disposed within a first time window 50 shown inFIG. 14. A second window sum I₂,3 is determined in association with asecond time window 52 and a third window sum I₃,4 is determined inassociation with a third time window 54 in the same manner as indicatedabove by summing the associated signal plus noise amplitudesA_(j).sup.(+) which are disposed within the second and third timewindows, respectively.

The window sums are transmitted uphole from the NMR tool 13 of FIGS. 1and 10 to the processing system or well logging truck computer 7adisposed on the wellbore surface as shown in FIG. 10. There are N_(w)window sums, where N_(w) is typically three to five in number.Therefore, since only N_(w) window sums are being transmitted upholeinstead of 2J amplitudes (there being R_(j) amplitudes and X_(j)amplitudes, where j =1, 2, 3, . . , J), the telemetry requirementsneeded to transmit the N_(w) window sums uphole to the truck computer7a, relative to the telemetry requirements needed to transmit the 2Jamplitudes uphole, is significantly reduced.

Referring again to FIG. 13, the second part 7a4A of the signalprocessing method and apparatus of the present invention is illustrated.

In FIG. 13, recall that the RMS noise (SQRT psi) is output from thefirst part 46a1 of the signal processing method and apparatus purposes:

1. to compute "gamma", block 4A1--the computation of "gamma" isdiscussed in connection with equation (42) of the Detailed Descriptionand is set forth in Appendix A of the Detailed Description of thePreferred Embodiment, entitled "An Algorithm for Optimal Selection ofgamma"; in Appendix A, note that the best value of "gamma" can be foundby finding the roots of equation (A.26) or of similarly derivedequations; and

2. to compute standard deviations "sigma(phi_(nmr))", "sigma(phi_(f))",and "sigma(T₂,log)", block 4A2--the standard deviation"sigma(phi_(nmr))" may be determined from equation (58) of the DetailedDescription of the preferred embodiment set forth below; the standarddeviation "sigma(phif)" may be determined from equation (61) of theDetailed Description; and the standard deviation "sigma(T₂,log)" may bedetermined from equation (65) of the Detailed Description set forthbelow.

The parameter "gamma", computed in block 4A1, is used to construct andminimize the likelihood function (-ln L), block 4A3. The likelihoodfunction (-ln L) is represented by the following equation, which isequation (42) of the Detailed Description set forth below: ##EQU3##

The spectral amplitudes (a₁) are determined by minimization of equation(42) subject to a positivity constraint, as indicated in the DetailedDescription set forth below in connection with equation 42.

The spectral amplitudes (a₁) are used for two purposes: to compute logoutputs "phi_(nmr) ", "phif", and T₂,log; and to compute signaldistributions P_(a) (log T₂) represented by color maps 2A1 of FIG. 16,to be discussed below.

To compute log outputs "phi_(nmr) ", "phi_(f) ", and T₂,log, block 4A4,the log output "phi_(nmr) " is determined from equation (43), asfollows: ##EQU4##

The log output "phi_(f) " is determined from equation (44), as follows:##EQU5##

The log output T₂,log is determined frcm equations (54) and (55), asfollows: ##EQU6##

To compute signal distributions P_(a) (log T₂) which represent colormaps 2A1 of FIG. 16, block 4A5, the computation of the signaldistributions P_(a) (log T₂) is performed using equation (50) asfollows: ##EQU7##

Referring to FIG. 16, a new output record medium 7a2A is illustrated.This new output record medium, a new log adapted to be given to a clientfor evaluation of the formation traversed by the wellbore, is generatedin response to the receipt of the following information:

1. the log outputs "phi_(nmr) ", "phi_(f) ", and T₂,1og of block 4A4;

2. the signal distributions for color maps P_(a) (log T₂) of block 4A5;and

3. the standard deviations "sigma(phi_(nmr))", "sigma(phi_(f))", and"sigma(T₂,log)" of block 4A2.

The new output record medium 7a2A of FIG. 16 records the following newdata or information presented in the form of logs as a function of depthin the wellbore (see element 2A2 in FIG. 16) and in the form of a colormap (see element 2A1 of FIG. 16);

1. signal phase 2A3 from block alA;

2. RMS noise estimate 2A4 from block alC;

3. free fluid standard deviation 2A5 from block 4A2;

4. porosity standard deviation 2A6 from block 4A2;

5. free fluid porosity 2A7 from block 4A4;

6. total NMR porosity 2A8 from block 4A4; and

7. color Map 2A1.

The new output record medium 7a2A is given to a customer or client forthe purpose of determining the presence or absence of undergrounddeposits of hydrocarbons and the reservoir quality of the formationtraversed by the wellbore 10 of FIG. 1.

A functional description of the operation of the signal processingmethod and apparatus in accordance with the present invention is setforth in the following paragraphs with reference to FIG. 16a and FIG. 13of the drawings.

In FIGS. 16a and 13, the RF antennas 18 and/or 21 measure the precessionof the protons in the pores 19a1 of the volume of investigation 19 ofFIG. 3 and generate spin-echo receiver voltage pulses similar to thespin-echo receiver voltage pulses "echo 1", "echo 2", and "echo 3"illustrated in FIG. 9b of the drawings. Phase sensitive detection (PSD)circuits disposed within the electronics cartridge 24 integrate each ofthe spin echo receiver voltage pulses over a time interval, and theintegrated signals are recorded as spin-echo inphase (R_(j)) andquadrature (X_(j)) amplitudes. The processor 46c of the downholecomputer 46 in the electronics cartridge 24 disposed within the NMR tool13 in the wellbore begins executing the first part 46a1 of the signalprocessing (software) method and apparatus of FIG. 13 stored within thememory 6a of the downhole computer 46. When the processor 46c completesthe execution of the first part of the signal processing software 46a1stored in memory 46a, the following data and information is determined:

1. the signal phase (theta) is estimated from the 2J inphase (R_(j)) andquadrature (X_(j)) amplitudes corresponding to J spin-echo receivervoltage pulses in the R and X channels using equation 9 as a function ofthe inphase and quadrature amplitudes set forth in the detaileddescription, block a1A of FIG. 13;

2. a spin-echo signal plus noise amplitude A_(j).sup.(+) is determinedfor each inphase (R_(j)) amplitude, quadrature (X_(j)) amplitude, andsignal phase (theta) using equation 22 in the Detailed Description; andthe amplitude A_(j).sup.(-) is also determined from inphase (R_(j))amplitude, quadrature (X_(j)) amplitude, and signal phase (theta) usingequation 23 in the Detailed Description, block alB of FIG. 13;furthermore, there are J inphase amplitudes (R_(j)) and J quadratureamplitudes (X_(j)); and, as a result, there are J spin-echo signal plusnoise amplitudes A_(j).sup.(+) and there are J amplitudes A_(j).sup.(-);

3. the N_(w) window sums I_(m),m+1 are determined from the A_(j).sup.(+) signal plus noise amplitudes using equation 35, where N_(w) istypically three to five in number, thereby reducing telemetryrequirements for transmission of window sums uphole to the surfacecomputer 7a, block a1D of FIG. 13; and

4. the RMS noise SQRT PSI is determined from the J amplitudesA_(j).sup.(-) using the practical implementation of equation 31 orequation per se, block a1C of FIG. 13.

Two sets of data are transmitted uphole from the NMR tool 13 to thesurface computer 7a: the N_(w) Window sums I_(m),m+1 and the RMS noiseSQRT PSI.

The processor 7a3 of the surface computer 7a receives the N_(w) windowsums I_(m),m+1 and the RMS noise SQRT PSI from the downhole computer 46of the NMR tool 13 disposed downhole; in response, the processor 7a3begins to execute the second part of the signal processing software 7a4Aof FIG. 13 stored in memory 7a4 of the surface computer 7a. When theprocessor 7a3 completes execution of the second part of the signalprocessing software 7a4A, the following additional data and informationis determined:

1. the standard deviations "sigma(phi_(nmr))", "sigma(phi_(f))", and"sigma(T₂,log) " are determined, block 4A2 of FIG. 13, the standarddeviation "sigma(phi_(nmr))" being determined from equation (58) of theDetailed Description, the standard deviation "sigma(phi_(f))" beingdetermined from equation (61) of the Detailed Description, and thestandard deviation "sigma(T₂,log)" being determined from equation (65)of the Detailed Description of the Preferred Embodiment set forth below.

2. the parameter "gamma" is determined from the RMS noise SQRT PSI,block 4Al of FIG. 13, in the manner described in Appendix A of theDetailed Description and in connection with equation 42 in the DetailedDescription;

3. once the parameter "gamma" is determined, a likelihood function (-lnL) is constructed and minimized, the likelihood function beingrepresented by equation 42 of the Detailed Description, which equationis a function of the parameter "gamma", block 4A3 of FIG. 13;

4. the spectral amplitudes {a₁ } are determined by minimization of thelikelihood function (-1n L) of equation 42, and the spectral amplitudes{a₁ } are used for two purposes: to compute log outputs "phi_(nmr) ","phi_(f) ", and T₂, log, block 4A4 of FIG. 13, and to compute the signaldistributions P_(a) (log T₂), which signal distributions are illustratedin the form of the color maps 2A1 of FIG. 16, block 4A5 of FIG. 13;

The processor 7a3 of FIG. 16a instructs the recorder 7a2 to generate thenew output record medium 7a2A of FIG. 16 using the aforementionedrecently determined standard deviations "sigma(phi_(nmr))","sigma(phi_(f))", and "sigma(T₂,log)"; the signal distributions P_(a)(log T₂); and the log outputs "phi_(nmr) ", "phi_(f) ", and T₂,log, thenew output record medium 7a2A of FIG. 16 displaying the following newinformation:

1. signal phase 2A3 from block a1A;

2. RMS noise estimate 2A4 from block a1C;

3. free fluid standard deviation 2A5 from block 4A2;

4. porosity standard deviation 2A6 from block 4A2;

5. free fluid porosity 2A7 from block 4A4;

6. total NMR porosity 2A8 from block 4A4; and

7. color map 2A1.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT Introduction

The development of Pulsed Nuclear Magnetic Resonance logging tools toacquire downhole spin-echo measurements in earth formations penetratedby boreholes is a new technology. The measurement principles and pulsesequences have been recently published.¹,2 This new logging technologyprovides detailed formation evaluation information previously obtainableonly from costly laboratory analysis of conventional core data. Thisinformation presently includes but is not limited to: (1) total NMRporosity (φ_(nmr)), (2) free-fluid porosity (φ_(f)) (i.e., part of totalporosity which is movable), (3) bound fluid porosity (φ_(b)), (4)spin-spin (T₂) relaxation time distributions which are related to poresize distributions in sandstones, and (5) continuous permeability logsin sandstones. To extract this information from the measured spin-echoesrequires a signal processing algorithm which is capable of providing anaccurate and repeatable spectral decomposition of the measured data. Thealgorithm must be efficient and robust for real time processing of dataas it is continuously acquired by a moving logging tool. This reportdescribes a new algorithm which statisfies these conditions and providesPNMT logs which agree well with conventional core and other log data.Other approaches to spectral decomposition of NMR data have beenreported³⁻⁶. Kenyon, et al.⁵ used the algorithm reported by Gallegos andSmith³ to compute T₁ distributions in rocks from laboratory NMRinversion recovery data. Latour, et al.⁶ used a computationallyexpensive singular value decomposition algorithm to compute relaxationtime distributions in rocks from laboratory NMR data.

It is well-known that sedimentary rocks have a distribution of poresizes. This results in NMR signals in rocks which decay with adistribution of relaxation times. Mathematically, the signal processingproblem is to determine the distribution functions from the measureddata (i.e., solve an inverse problem). The aforementioned formationevaluation parameters are computed from the distribution functions.Generally, there exists a joint distribution function, i.e., a functionof both the longitudinal (T₁) and transverse (T₂) spin relaxation times,however, laboratory measurements have shown that these distributions insedimentary rocks are correlated.⁶ The correlation is valid at NMRfrequencies in the range of 2 MHz and is applicable to the boreholemeasurements described in this report.

Pulse Sequence and Parameters

It suffices to describe the pulse sequence and typical measurementparameters used in practice. The spin-echo measurement employed is thewell-known Carr-Purcell-Meiboom-Gill (CPMG) sequence. This sequencemeasures the decay of the amplitude of the transverse magnetizationfollowing a 90° (+x) r.f. pulse which rotates the magnetization into aplane transverse to an applied static magnetic field (H₀ z). Thefrequency of the r.f. pulses is equal to the median Larmor protonprecession frequency. The nuclear spins contributing to the transversemagnetization prcess at different Larmor frequencies in theinhomogeneous d.c. magnetic field. The free-induction decay (FID) signalwhich is generated by the 90° (+x) pulse decays to zero withinmicroseconds because of spin dephasing produced by the spread in Larmorfrequencies. The FID signal occus too soon following the 90° (+x) to bemeasured by the PNMT electronics. At times t_(j) =(2j-1)t_(cp) followingthe 90° (+x) pulse, a set of 180° (y) r.f. pulses are applied whichcause the transverse magnetization to refocus at times t_(j) =2jt.sub.cp for j=1,2, . . . , J producing J spin-echo signals. Note that ther.f. magnetic field of the 180° pulses is transverse to both H₀ and ther.f. magnetic field of the 90° pulses. The spin-echoes are equallyspaced in time with spacing Δ=2t_(cp). The train of spin-echoesrepresents a signal whose decay contains contributions from allcomponents of the intrinsic T₂ -distribution. Here intrinsic refers to adistribution that includes effects of microscopic spin-spin interactionsas observed in bulk liquids as well as surface relaxation from theconfining pores. It is the latter effects that frequently dominate inreservoir rocks and provide the link between T₂ -distributions and poresize distributions. The effects, on the spin-echo decay, of moleculardiffusion in the static field gradient can be made negligble for thePNMT by making the Carr-Purcell time t_(cp) sufficiently short. The CPMGsequence described above can be improved by using phase alternated pairs(PAPS) of CPMGs to eliminate baseline shifts. Phase alternated pairs ofCPMGs differ by shifting the phase of the 90° pulses 180° degrees. ThePNMT PAPS pulse sequences can therefore be written succinctly in anobvious notation.

    CPMG.sup.(±) :W-90°(±x)-(t.sub.cp -180°(y)-t.sub.cp -(echo).sub.j),.sub.j=1,2, . . . , J                      (1)

where W is a wait time that must precede each CPMG so that thelongitudinal magnetization can recover prior to the initial 90° pulse.The PAPS pairs are combined by taking one-half their difference. Thiseliminates baseline offsets. Note each PAPS is constructed by signalaveraging two CPMG sequences which reduces the rms noise by a factor √2.

Typical values of the pulse parameters are t_(cp) =100-500 μs, W=0.5-1.5s, and J=100-1000 echoes.

Statement of The Signal Processing Problem

The spin-echo amplitudes are obtained by hardware integration of thereceiver voltages over J time windows each centered about t_(j) =jΔ forj=1,2, . . . , J. The PNMT tool uses phase sensitive detection tomeasure the in-phase (R_(j)) and quadrature (X_(j)) components of thesignal-plus-noise amplitudes. As shown in the next section, the signalphase θ is estimated from the R_(j) and X_(j) amplitudes which are thencombined to provide the signal-plus-noise amplitudes A_(j).sup.(+). Aphenomenological model for the signal-plus-noise amplitude of the j-thecho can be written in the form: ##EQU8## where P_(j) (T₁,T₂) is thejoint relaxation time distribution function and the integals are overthe domain of the distribution function. As noted previously,experiments have shown that to within a good approximation, the jointdistribution function in reservoir rocks has the form,

    P.sub.J (T.sub.1,T.sub.2)=P(T.sub.2)δ(T.sub.1 -ξT.sub.2), (3)

where ξ˜1.5. That is, the T₁ and T₂ distributions are practicallyidentical except for a constant scaling factor. Substitution of eq. (3)into (2) and performing the integration over T₁ leads to ##EQU9## wherefor A_(j).sup.(+) expressed in volts, P(T₂)dT₂ is the signal amplitudein volts contributed by spectral components with transverse relaxationtimes in the interval T₂ to T₂ +dT₂. The domain of the function P(T₂) isthe closed interval [T_(min), T_(max) ]. The function f(W,ξT₂) accountsfor the incomplete recovery of the longitudinal magnetization during thewait time W and is defined by, ##EQU10##

The integrals in eqs. (2) and (4) are the amplitudes of the transversemagnetization observed at the j-th echo and N_(j).sup.(+) is the thermalnoise in the tool electronics.

Equation (4) is a Fredholm integral equation of the first kind for thedistribution function P(T₂). The solution of eq. (4) for P(T₂) from themeasured spin-echo signal-plus-noise amplitudes is the signal processingproblem that must be confronted to obtain maximum information from thePNMT data. This type of problem represents an inverse problem of thetype frequently encountered in remote sensing problems.⁷ The relaxationtime distribution is the central quantity of interest since essentiallyall of the petrophysical quantities of interest can be computed fromthis function.

Pre-Processing of Spin-Echo Data

As noted previously, the amplitudes of the spin-echoes are integratedover time windows and the integrated signals are recorded as R_(j) andX_(j) time series channels or waveforms. Each time series orCPMG.sup.(+) is combined with its phase alternated pair CPMG(-). ThePAPS pairs are accumulated as the tool moves and then averaged andoutput into depth bins. Thus, in each depth bin the data typicallyconsists of several hundred echoes R_(j) and X_(j) for j=1,2, . . . , Jwhere J is the total number of echoes collected. Prior to applying thesignal processing algorithm, the data in each depth bin arepre-processed. First, an estimate θ of the signal phase is computed.Using θ the R_(j) and X_(j) data are combined into two random timeseries A_(j).sup.(+) and A_(j).sup.(-). The random variablesA_(j).sup.(+) have the statistical properties of the phase coherentsignal-plus-noise and are used to estimate the relaxation timedistribution (i.e., by solving a discretized version of eq. (4)). Therandom variables A_(j).sup.(-) have the statistical properties of thenoise and are used to estimate the rms noise, √ψ, on a single echo ofthe PAPs pairs in each depth bin. In each depth bin the number of PAPspairs accumulated and averaged depends on the pulse sequence parametersand the logging speed.

Statistical Properties of the Noise

The spin-echo in-phase and quadrature amplitudes can be written in theform,

    R.sub.j =S.sub.j cosθ+N.sub.j.sup.(c),               (6)

    X.sub.j =S.sub.j sinθ+N.sub.j.sup.(s),               (7)

where θ is the signal phase and S_(j) is the signal. N_(j).sup.(s) arethermal noise voltage fluctuations in the R and X receiver channels,respectively. The thermal noise fluctuations are uncorrelated, zero meanGaussian random variables with the following statistical properties:

    <N.sub.j.sup.(c) >=<N.sub.j.sup.(s) >=0,                   (8a)

    <N.sub.j.sup.(c) N.sub.k.sup.(s) >=0,                      (8b)

    <N.sub.j.sup.(c) N.sub.k.sup.(c) >=<N.sub.j.sup.(s) N.sub.k.sup.(s) >=δ.sub.j,k ψ.                                  (8c)

The angular brackets are used to denote statistical or ensemble averagesand δ_(j),k is the Kronecker delta function. Note that thermal noisefluctuations are a time translationally invariant stochastic process.Therefore it follows from the ergodic therorem taht statistical averagescan be replaced by time averages. In eq. (8c), ψ is the noise varianceon a single echo.

Signal Phase Estimator

An unbiased estimator θ of the signal phase is computed from thein-phase and quadrature amplitudes, ##EQU11##

To compute the mean and variance of θ, sum Eqs. (6) and (7) over allechoes, i.e., ##EQU12## from which it follows that, ##EQU13## where Ihave defined, ##EQU14##

Combining eqs. (12) and (13) one finds to lowest order in the noisefluctuations that, ##EQU15## from which is follows that ##EQU16## wherein obtaining the above equation I have used the result, ##EQU17## whichis valid for θ≃θ. It follows easily from eq. (17) and the statisticalproperties of the noise that,

    <θ>=θ,                                         (19)

so that θ is an unbiased estimator of the signal phase θ. One also findsusing the noise properties that, ##EQU18##

Finally, it follows from eqs. (19) and (20) that the variance in thephase estimator θ is given by, ##EQU19##

It should be noted that the phase estimator θ is sensible provided thatthe signal-to-noise ratio is not too small. In practice, the signalphase is found to be relatively constant in zones with porositiesgreater than a few p.u. and exhibits random fluctuations in zones withno appreciable signal. It is a useful tool diagnostic since it shouldnot vary with the formation and should be relatively constant in porouszones during a logging run.

Computation of A_(j).sup.(+) and A_(j).sup.(-) :

Using θ, it is convenient to construct two random time series from theR_(j) and X_(j). These are the signal-plus-noise amplitudesA_(j).sup.(+) introduced in eq. (2) and A_(j).sup.(-) which can be usedto estimate ψ from the data. These random time series are defined by,

    A.sub.j.sup.(+) =R.sub.j cosθ+X.sub.j sinθ,    (22)

    and

    A.sub.j.sup.(-) =R.sub.j sinθ-X.sub.j cosθ.    (23)

The rationale for introducing A_(j).sup.(+) is that it has thestatistical properties of a phase coherent signal. The above pair ofequations has a simple vectorial interpretation, e.g. in eq. (22) thetwo terms on the right are the projections of the R and X componentsalong the total signal whereas the terms in eq. (23) are projectionsperpendicular to the total signal. To calculate its expectation valuefirst note that,

    A.sub.j.sup.(+) =S.sub.j cos(θ-θ)+N.sub.j.sup.(+) ≃S.sub.j +N.sub.j.sup.(+),                  (24)

where I have used eqs. (6), (7) and (22) and defined,

    N.sub.j.sup.(+) =N.sub.j.sup.(c) cosθ+N.sub.j.sup.(s) sinθ. (25)

I have dropped terms in eq. (24) of order (θ-θ)² which are assumed to benegligble. The expectation value of eq. (24) is obtained from thestatistical properties of the noise, i.e., eqs. (8).

    <A.sub.j.sup.(+) >≃S.sub.j.                  (26)

One also finds that,

    <<A.sub.j.sup.(+)).sup.2 >≃S.sub.j.sup.2 +ψ, (27)

from which it follows that the variance in A_(j).sup.(+) is ψ. I shallmake use of these results in the development of the algorithm forcomputing the spectral distrubtion function. Next it is shown thatA_(j).sup.(-) has the properties of the noise. Substituting eqs. (6),(7) into (23) one finds that,

    A.sub.j.sup.(-) =S.sub.j sin(θ-θ)+N.sub.j.sup.(-) ≃N.sub.j.sup.(-),                           (28)

    where

    N.sub.j.sup.(-) =N.sub.j.sup.(c) sinθ-N.sub.j.sup.(s) cosθ. (29)

Using the statistical properties of the noise one finds that,

    <(A.sub.j.sup.(-) >≃0,                       (30)

    and,

    <<A.sub.j.sup.(-)).sup.2 >≃ψ,            (31)

from which it follows that the variance in A_(j).sup.(-) is ψ. Thus, asnoted previously, the random time series for A_(j).sup.(-) can be usedto estimate the rms noise √ψ.

Spectral Decomposition Algorithm Discretiztaion of the Signal

First, the signal S_(j) which is given by the integral in eq. (4) isdiscretized, i.e., ##EQU20## where it is assumed that there are N₃ ,components (i.e., basis functions) in the spectrum with relaxation timesgiven by, ##EQU21## for l=1,2, . . . , N₃. Note that the T₂,l areequally spaced on a logarithmic scale. In Eq. (32), I have defined thepolarization functions, ##EQU22##

The signal processing problem therefore becomes the determination of theN₃ spectral amplitudes α_(l) ≡P(T₂,l)δ_(l) where δ_(l) =(T₂,l+1-T₂,l-1)/2.

Data Compression

In this section, the data A_(j).sup.(+) are reduced by introducing sumsof echoes over time windows. As noted earlier, the different windowshave different sensitivites to the various components of the spectrum.Consider N_(w) non-overlapping windows. Define the window sum,I_(m),m+1, over (N_(m+1) -N_(m) +δ_(m),1) echoes in the m-th window,i.e., ##EQU23## where ρ_(m) =(1-δ_(m),1) for m=1,2, . . . N.sub.ω andδ_(m),1 is the familiar Kronecker delta function (e.g., equals 1 for m=1and 0 otherwise). Here (N_(m) +P_(m)) and N_(m+1) are the echo numbers(i.e., endpoints) of the m-th window. More explicitly, for the firstwindow ##EQU24## where N₁ and N₂ are the echo numbers of the endpointsof the first window. For the second window, ##EQU25## and, in general,for the n-th window where n>1, ##EQU26##

Recalling eq. (32), one can write (35) in the form, ##EQU27## where Ihave defined window spectral sensitivity functions F_(m),m+1 given bygeometric series, i.e., ##EQU28## and also defined the sums over noise,##EQU29##

In each window the set of sensistivity functions defined above for l=1,. . . N, provide the relative sensitivity of each window to thedifferent components in the spectrum.

Performing the summation in eq. (37), one finds that the sensitivityfunctions can be expressed in a form convenient for computation, i.e.,##EQU30## where x₁ =Δ/T₂,1. In practice, it has been found that for PNMTlog data that usually 3 windows are sufficient (i.e., the log outputsare not altered by adding more windows). Likewise for continuous loggingit has been found that N_(s) =8 is sufficient. This is discussed furtherin a section that examines the statistical uncertainties in the logoutputs.

Statistical Properties of Window Sums

The statistical properties of the reduced set of random variableI_(m),m+1 are determined from the known statistical properties of thenoise. One finds from eq. (36) that their expectation values are givenby, ##EQU31## where the curly brackets on the right are used to indicatethat the expectation value is a functional (i.e., a scalar whose valuedepends on a function) of the spectral amplitudes. The above expressionhas a simple physical interpretation, i.e., the expected value of thesum of the signal over the m-th window is the weighted sum of thesensitivity functions of the various components each weighted by itsamplitude. The variances in the window sums are easily calculated. Onefinds that,

    σ.sup.2 (I.sub.m,m+1)=ψ(N.sub.m+1 -N.sub.m +δ.sub.m,1)≡ψσ.sub.m,m+1.           (41)

Here ψ is the variance of the noise in a single echo which is the samefor all echoes. The variance of the window sum in the above equation issimply the number of echoes in the window times the variance in a singleecho. Note that this result is evident from elementary statistics sincethe variance in a sum of uncorrelated random variables is simply the sumof the variances. For non-overlapping windows the window sums overdifferent windows are uncorrelated, i.e.,

    <δI.sub.m,m+1 δI.sub.m,m+1 >=δ.sub.m,m ψσ.sub.m,m+1.sup.2,                             (41a)

where from eq. (36), δI_(m),m+1 =N_(m),m+1.

Spectral Estimation

The window sums defined in eq. (35) are independent Gaussian randomvariables with expectation values and variances given by eqs. (40) and(41), respectively. The logarithm of the likelihood function for theN.sub.ψ window sums is given by, ##EQU32## where the expected values<I_(m),m+1 >≡I_(m),m+1{a₁ } were defined earlier and are functions ofthe spectral parameters which we want to estimate. The last term is aphenomenological regularization term introduced to prevent noiseamplification artifacts. It is a penalty constraint that prevents the L₂norm of the amplitudes from becoming too large. The L₂ regulatization iscommonly employed but other criteria can be imposed. The spectralamplitudes are relatively insensitive to the dimensionless parameter δ.A value γ≃5 is usually used for PNMT log data. How to best choose γ isnot a trivial question. The best fit to the data occurs if γ=0, however,the solution is not stable. It would be the best solution in the absenceof noise. In the presence of noise, the squared residuals of the fit ofthe data in each window should be eaual to the variance in the windowsums in eq. (41). This represents the best fit based on the statisticsof the model. Increasing γ reduces the variance in the estimates but canlead to solutions that do not fit the data if γ is too large. Analgorithm for a priori estimation of an "optimal" γ determined at eachdepth from the data is developed in Appendix A. The spectral amplitudes{a₁ } are determined by minimization of eq. (42) subject to a positivityconstraint. The estimation is extremely fast on a computer as only a few(e.g., 3) windows are needed. The computation time is essentiallyindependent of the number of echoes.

The above algorithm leads to a tremendous data reduction since thespectrum is obtained from only a few random variables instead ofhundreds or thousands of echoes. This huge data reduction has obviouspotential benefits for efficient data acquisition and storage. A set(e.g., 10-100) of window sums can be rapidly computed downhole andtransmitted uphole for processing. This set can be combined into gatesfor uphole processing. This leads to a substantial reduction in PNMTtelemetry requirements which is important for commercial tools run incombination. Downhole preprocessing can be used to access data qualityand flags established for sending all of the echoes uphole if necessary.

Total, Free and Bound Fluid Porosities

The spectral analysis described above leads immediately to logs oftotal, bound and free fluid porosities and mean transverse spinrelaxation time. The total NMR porosity φ_(nmr) can be computed byintegration of the distribution function P(T₂) defined in eq. (4), i.e.,##EQU33## where K_(tool) is a tool constant for converting volts toporosity. Similarly, the free-fluid porosity is given by, ##EQU34##where for the free fluid porosity (denoted by UBF on PNMT logs), thesummation is over a subset of components, I=N_(c), N_(c) +1, . . . N_(s)for which T₂,1 ≧T_(c). Centrifuge experiments by Straley, et al.⁸ havefound that in many sandtones T_(c) ≃33 milliseconds. The bound-fluidporosity is simply given by φ_(bf) =φ-φ_(f). The term Δφ is a smallcorrection which accounts for the fact that the cut-off T_(c) does notlie on the endpoint of the free fluid integration interval. An analyticexpression is derived in Appendix C for the Δφ correction. Note thatthis correction does not affect φ_(t), i.e., only the partitioning ofthe free and bound fluid porosities.

Distribution And Mean Relazation Times

The porosity distribution function P(T₃) is with respect to T₂. Fordisplaying maps of porosity distributions versus relaxation time, it isuseful to define a logarithmic distribution P_(a) (logT₂) since therelaxation times span several decades. In terms of the logarithmicdistribution function, ##EQU35## where K_(tool) P_(a) (logT₂)d(logT₂) isthe porosity in the interval [logT₂,logT₂ +d(logT₂)]. The distributionwith respect to T₂ and logT₂ are related, i.e., ##EQU36## withc=(ln10)⁻¹ as can be shown using eqs. (43) and (45). Using thediscretization of P(T₂) introduced earlier, the discretized distributionwith respect to the T₂,1 defined in eq. (33) is given by, ##EQU37## withδ₁ =(T₂,1+1 -T₂,1-1)/2. Using eq. (33), it follows from simple algebrathat, ##EQU38## where I have defined, ##EQU39## Combining eqs.(46)-(48), one finds that, ##EQU40## Note that to within a constantfactor, independent of l, the logarithmic distribution is proportionalto the amplitude distribution {a₁ }.

The main results concerning distributions are eqs. (47) and (50). Insummary, to display the distribution P(T₂), use eq. (47) to computeP(T₂,1) and plot it versus T₂,1 on a linear scale. To display, thelogarithmic distribution use eq. (50) to compute P_(a) (logT₂,1) andplot it versus T₂,1 on a logarithmic scale. More simply, plot the {a₁ }distribution on a logarithmic T₂,1 scale to obtain the logarithmicdistributin to within a constant scale factor.

A mean relaxation time T₂ can be defined for the distribution P(T₂).That is, ##EQU41## or in discretized form, ##EQU42##

Analagously, a logarithmic mean relaxation time T₂,log can be defined.First, one defines a mean logarithm m for the P_(a) (logT₂)distribution, i.e., ##EQU43## or in discreted form, ##EQU44## Note thaton a logarithmic scale the spacings, δ=log T₂,l+1 -log T₂,l ≡(N_(s)-1)⁻¹ log[T_(max) /T_(min) ], are equal and therefore independent of l.The last equality in the above equation follows from the result in eq.(50). The logarithmic means relaxation time T₂,log is obtained byexponentiation,

    T.sub.2,log =10.sup.m.                                     (55)

The logarithmic means relaxation times T₂,log and porosities φ_(nmr) areused as inputs into empirically derivatived equations to provideestimates of permeabilities.

Variances In The Porosity Estimators

The variances in the porosity estimators φ_(nmr), φ_(f) and φ_(bf) arecomputed from the covariance matrix C_(l),k. The porosity estimators areobtained from estimates of the spectral amplitudes, i.e., ##EQU45##where the hat is used to differentiate the estimators from the truequantities (e.g., in eqs. (43)-(44) true porosities and spectralamplitudes are indicated). The variance σ² (φ_(nmr)) is by definition,

    σ.sup.2 (φ.sub.nmr)≡<φ.sub.nmr.sup.2 >-(<φ.sub.nmr >).sup.2,                                                 (57)

or explicity by, ##EQU46## where the parameter-parameter covariancematrix,

    C.sub.l,k =<δa.sub.l δa.sub.k >,               (59)

has been introduced. The angular brackets denote statistical averagesand the fluctuations δa_(k) are defined by,

    δa.sub.k =a.sub.k -<a.sub.k >.                       (60)

Note the covariance matrix is symmetric and its diagonal elements arethe variances in the amplitudes, i.e., σ² (a_(l)). In general, thefluctuations in the amplitude estimates are correlated and therefore theoff-diagonal elements of C_(l),k are non-zero. The derivation of thevariances in the free and bound fluid porosities is analogous to thederivation of eq. (58). One finds, for example, that ##EQU47## whereN_(c) is defined in eq. (44). In order to apply eps (58) and (61), oneneeds to compute the convariance matrix. An approximation for thecovariance matrix will be derived in a subsequent section. First,however, an approximate calcuation of the variance in the logarithmicmean relaxation time is presented.

Variance In The Logarithmic Mean Relaxation Times

In this section an approximate formula for the variance T₂,log isderived. Recalling eqs. (54) and (55) one finds that, ##EQU48## wherethe curly bracket on the left indicates that T₂,log is a functional ofthe amplitude estimates a_(l). It is to be understood that thesummations on the right are over the whole spectrum, i.e., l=1, 2, . . .N_(s). The constant c=(ln 10)⁻¹. Expand T₂,log in a Taylor's seriesabout the expectation values of the amplitude estimates, i.e., ##EQU49##Note that in eq. (63), it has been assumed that the fluctuations aresufficiently small that terms of order (δa_(k))² can be neglected. Thederivatives on the right are, of course, evaluated at <a_(k) >. Toproceed, another small fluctuation approximation, i.e., <T₂,log >≃T₂,log(<a_(l) >) is employed to obtain ##EQU50## where, δT₂,log ≡(T₂,log-<T₂,log >), and eqs. (59) and (63) have been used. The partialderivatives can be evaluated explicitly using eq. (62). One finds aftersome simple algebra that the standard deviation, ##EQU51## where I havedefined the quantities, ##EQU52## where eq. (33) was used in obtainingthe above result. Here δ is the T₂,l spacing on a logarithmic scale(e.g., see the remarks following eq. (54)). In actual computations, theexpectation values in the above equations are replaced by the maximumlikelihood estimates obtained by the minimization of eq. (42). Thecalculation of the variance in T₂ can be derived by similarmanipulations but is not given here. The notion of a relaxation timebecomes meaningless whenever the signal is dominated by the noise. Thisoccurs in low porosity (e.g., for φ_(nmr) ≃1 p.u.) formations and leadsto random fluctuations in T₂,log usually occuring at long times sincenoise amplitude fluctuations are non-decaying. In these instances eq.(65) provides a useful criterion for "turning off" the T₂,log log curve.A criterion which has proven useful is to disallow the log curve if thefactor multiplying T₂,log on the right side of eq. (65) exceeds unity.

Calculation of the Parameter-Parameter Covariance Matrix

An exact covariance matrix can be calculated for the algorithm. Usingeqs. (A.1)-(A.3) and eq. (42), it is easy to prove that the parameterestimates are related to the "data" via the set of linear equations,

    a.sub.l =R.sub.l,m I.sub.m,m-1,                            (67)

for l=1, 2, . . . , N_(s) and where the Einstein summation convention ofsumming over repeated indices is used in this section. In the aboveequation, I have defined the N_(s) ×N_(w) matrix R,

    R.sub.l,m =M.sub.l,k.sup.-1 Q.sub.k,m,                     (68)

where the N_(s) ×N_(s) matrix M is defined by, ##EQU53## and where theN_(s) ×N_(w) matrix Q is defined by, ##EQU54## where δ_(m),m+1 isdefined in eq. (41). Using the definition of the parameter-parametercovariance matrix in eq. (59) and eqs. (41) and (41a) one finds that,

    C.sub.l,k =R.sub.l,m ψσ.sub.m,m+1.sup.2 R.sub.m,k.sup.t. (71)

The parameter-parameter convariance matrix in eq. (71) can be used witheqs. (58), (61) and with an analogous equation for σ² (φ_(bf)) to studythe standard deviations in φ_(t), φ_(f) and φ_(bf) for various pulse andprocessing parameters. Some of these results are shown in FIGS. 17-20.The PNMT logs from repeat runs generally agree very well with thestandard deviations computed from the covariance matrix. It should benoted, however, that log repeatability can be affected by many factorsother than statistical fluctuations (e.g., hole conditions). Thereforethe uncertainties computed using eq. (71) and displayed on the logs mayin some cases be optimistic compared to the log uncertainties estimatedfrom statistical analysis of repeat runs. A proof that the matrix M ispositive definite and therefore has an in Appendix B.

Processing Example From a PNMT Field Test

The algorithm described above has been used to process log data fromfive Schlumberger clients wells logged to date. A flowchart illustratingthe various steps in the implementation of the algorithm in shown inFIG. 13.

A detailed discussion of the field examples is beyond the scope of thisreport. Here, it will suffice to show a short section of processedcontinuous PNMT log data and the analysis of station data acquired atseveral depths within the interval. The log data shown was acquired withthe PNMT tool moving at 150 ft/hour. FIG. 16 shows a section of log froma well in Texas. The section shown contains a non-hydrocarbon bearingthinly bedded sand/shale sequence. In Track 1, a color map of signalversus relaxation time (T₂) plotted on a logarithmic scale is shown. Themagnitude of the signal is proportional to the intensity of the colorand in this example, the T₂ range is from 1 ms to 1500 ms. The red curvein Track 1 is the logarithmic means (T₂,log) of the distribution. InTrack 2, the rms noise estimate (√ψ) in volts, the estimated standarddeviations in the total and free-fluid porosities, and the signal phaseestimate (θ) in degrees are displayed. Note that θ is essentiallyconstant, as expected, except at depths with low porosities (i.e., lowsignals) where the phase estimator is dominated by random noise. InTrack 3, the total and free-fluid porosity estimates are displayed.

Several station stop measurements were made in the logged interval. Atstation stops, the tool acquires data for several minutes. The data isaveraged to reduce the noise so that the quality of the station data issignificantly better than the continuous data. The signal processing ofthe station data is the same as for the continous data.

In FIG. 21, the signal distribution (analogous to the color map shown inFIG. 16) versus T₂ is shown for a station stop at 1100 ft. Recall thatthe area under the signal distribution curve is proportional toporosity. Comparing FIG. 21 with the continuous log observe that thetotal NMR porosity (φ_(nmr)), free-fluid porosity (φ_(f)), andlogarithmic means relaxation time (T₂,log) computed from the stationstop data agree well with the continuous log values. The values shownwere computed with a regularization parameter, γ=5.0, as was thecontinuous log. It should be noted, that it has been found thatestimates of φ_(nmr), φ_(f) and T₂,log depend only weakly on theregulariation parameter. For example, changing γ by an order ofmagnitude usually results in porosity variations of less than ±0.5 pu.The detailed shape of the signal distribution, however, can be changedsignificantly in some cases by varying the regulation parameter. This isa consequence of the fact that there are infinitely many solutions whichwill fit the data. Decreasing γ reduces the fit error but can increasethe norm of the solution vector.

In FIG. 21, the two signal distributions plotted correspond to two verydifferent values of the regularization parameter. Note that there ispracticially no difference in the two distributions. Both distributionsfit the data to within the noise and have comparable norms. Thereforeboth solutions are mathematically acceptable.

In FIG. 22, results from a station stop at 1125 ft are displayed. Thetwo signal distributions corresponding to values of γ differing by anorder of magnitude are qualitatively similar. The distribution computedwith γ=0.5 amplifies the two peak structure already apparent in thedistribution computed using γ=5.0. In this example, the two peaks areprobably due to signal contributions from two disparate poredistributions in the thinly bedded heterogeneous reservoir at 1125 ft asindicated on the FMS image (not shown here). In oil reservoirs, wherethe formation is relatively homogeneous over the length of the PNMT toolaperture, separate oil and water signal peaks can be identified in thesignal distribution.

In FIG. 23, results from a station stop at 1148 ft are shown. Note that,the continuous log outputs agree will with those obtained by processingthe station data. Note that, the reservoir quality at this depth is poorcompared to that at the previous two stations as evidenced by almost allof the signal being associated with bound fluid porosity.

In FIGS. 24 and 25 the station stops at 1200 ft and 1229 ft are shown.These distributions, reveal relatively high permeability reservoir rockat these depths. The long relaxation times are indicative of large poreand, pore surfaces free of iron or other magnetic material.

References

The following references are incorporated into this specification byreference:

1. Sezginer, A., Kleinberg, R. L., Fukuhara, A., and Latour, L. L.,"Very Rapid Measurement of Nuclear Magnetic Resonance Spin-LatticeRelaxation Time and Spin-Spin Relaxation Time," J. of MagneticResonance, v. 92, 504-527 (1992).

2. Kleinberg, R. L., Sezginer, A., Griffin, D. D., and Fukuhara, M.,"Novel NMR Apparatus for Investigating an External Sample," J. ofMagnetic Resonance, v. 97, 466-485 (1992).

3. Gallegos, D. P. and Smith, D. M., "A NMR Technique for the Analysisof Pore Structure: Determination of Continuous Pore Size Distributions,"J. of Colloid and Interface Science, v. 122, No. 1, pp. 143-153, Mar.,1988.

4. Brown, R. J. S., Borgia, G. C., Fantazzini, P., and Mesini, E.,"Problems In Identifying Multimodal Distributions Of Relaxation TimesFor NMR In Porous Media," Magnetic Resonance Imaging, Vol. 9, pp.687-693 (1991).

5. Kenyon, W. E., Howard, J. J., Sezginer, A., Straley, C., Matteson,A., Horkowitz, R., and Erlich, R., "Pore-size Distribution and NMR inMicroporous Cherty Sandstones," Trans. of the SPWLA of the 30th Ann.Logging Symp., Paper LL, Jun. 11-14, 1989.

6. Latour, L. L., Kleinberg, R. L., and A. Sezginer, "Nuclear MagneticResonance Properties of Rocks at Elevated Temperatures," J. of Colloidand Interface Science, v. 150, 535 (1992).

7. Twomey, S., "Introduction To The Mathematics of Inversion In RemoteSensing And Indirect Measurements," published by Elsevier ScientificPublishing Company, 1977.

8. Straley, C., Morriss, C. E., Kenyon, W. E., and Howard, J. J., "NMRin Partially Saturated Rocks: Laboratory Insights on Free Fluid Indexand Comparison with Borehole Logs," Trans. of the SPWLA 32nd Ann.Logging Symp., Paper CC, Jun. 16-19, 1991.

9. Butler, J. P., Reeds, J. A. and Dawson, S. V., "Estimating Solutionsof First Kind Integral Equations with Non-Negative Constraints andOptimal Smoothing," SIAM J. Numerical Anal., v. 18, No. 3, 381-397,1981.

APPENDIX A: AN ALGORITHM FOR OPTIMAL SELECTION OF γ

In this Appendix an algorithm for selection of an "optimal"regularization parameter γ_(opt) is derived. A similar algorithm wasderived by Butler, Reeds and Dawson⁹ and has been used by Gallegose andSmith³ to select the regularization parameter. The selection criteionfor choosing γ_(opt) is based on the notion that an optimal value ofsmoothing is obtained by minimizing the squared norm of the differencevector δa=a.sub.γ -a_(true), where a.sub.γ is the regularized solutionand a_(true) is the true distribution. Since a_(true) is, of courseunknown, a compromise is made by replacing a_(true) by a hypotheticalnose free solution a⁰ for which γ=0. Although, the aforementionedcriterion can be rigorously stated, assumptions must be made thatintroduce elements of empiricism into the algorithm. These elements arealso present in other algorithms⁹ and therefore the claims of optimallyare somewhat subjective.

Maximum likelihood estimates of the N_(s) spectral amplitudes a_(l) areobtained by minimization of, -ln L, in eq. (42). The equations to besolved are ##EQU55## for l=1, 2, . . . N_(s). Substituting eq. (42) into(A.1) leads to the linear system of equations, ##EQU56## where thesymmetric matrix M_(l),k is defined in eq. (69) and the data vector##EQU57## has been defined. The quantities f_(l), I_(m),m+1 andF_(m),m+1 (T₂,l) are defined in eqs. (34), (35) and (39), respectively,and

    σ.sub.m,m+1.sup.2 =N.sub.m-1 -N.sub.m +δ.sub.m,1, (A.4)

have been defined. The σ_(m),m+1² are simply the number of echoes in them-th window.

It is useful to write eq. (A.2) in an explicity matrix form,

    M a.sub.γ =d,                                        (A.5)

and to also write the companison equation,

    M.sub.0 a.sup.(0) =d.sup.(0),                              (A.6)

where a.sub.γ is a vector whose N_(s) components are the spectralamplitudes for a regularization parameter γ, d is the data vectordefined in eq. (A.3). The matrix M₀ is defined by the matrix equation,

    M=M.sub.0 +γI                                        (A.7)

where IεR^(N).sbsp.s.sup.×N.sbsp.s is the identity matrix. In (A.6),a.sup.(0) is the vector of spectral amplitudes corresponding tohypothetical noise free data and d.sup.(0) is the noise free datavector, i.e.,

    d=d.sup.(0) +n,                                            (A.8)

where the noise vector n is defined by, ##EQU58## where N_(m),m+1 isdefined in eq. (38).

It follows from the theory of matrices that the real symmetric matrix Mcan be diagonalized by an orthogonal transformation, i.e.,

    M=U D.sub.γ U.sup.t,                                 (A.10)

where U^(t) is the transpose of U where U is an N_(s) ×N_(s) matrixwhose columns are an orthonormal set of eigenvectors of M, i.e., thevectors u_(j) satisfy the eigenvalue equation,

    M u.sub.j =λ.sub.j (γ)u.sub.j,                (A.11)

and the othonormality conditions, ##EQU59## or the matrix equivalent,

    U.sup.t U=I.                                               (A.12b)

where λ_(j) (γ) is the eigenvalue of M associated with eigenvectoru_(j). The N_(s) ×N_(s) matrix D.sub.γ in eq. (A.10) is a diagonalmatrix with the eigenvalues λ_(j) (γ) on the diagonal. The aboveequations result from the orthonormality of the columns of the matrix M.Note that the operator M does not have a null space, i.e., it ispositive definite for γ>0.

It can also be proven for orthogonal transformation square matrices likeU that the rows are likewise orthonormal which leads to the equations,##EQU60## or its matrix equivalent,

    U U.sup.t =I.                                              (A.13b)

The operator M₀ has a non-trivial null space because the rank (denotedby r) of M₀, i.e., the dimension of its non-null space is less thanN_(s). M₀ has reduced rank because the data are not independent. This istypical of most inversion problems and mathematically the problem isunderdetermined (more unknowns than measurements). In mathematicalterms,

    M.sub.0 u.sub.j =λ.sub.j (0)u.sub.j,                (A.14a)

for j=1,2, . . . , r where the eigenvalues can be ordered so that λ₁(0)>λ₂ (0)>. . . >λ_(r) (0)>0. Also in the null space,

    M.sub.0 u.sub.j =0,                                        (A.14b)

for j=r+1, . . . , N_(s). The operator M₀ can be diagonalized in itsnon-null space by the transformation,

    M.sub.0 =U.sub.r D.sub.0 U.sub.r.sup.t,                    (A.15)

where U_(r) is an N_(s) ×r matrix whose columns are the eigenvectorsu_(j) where j=1, 2, . . . , r, and D₀ is a diagonal matrix witheigenvalues λ_(j) (0) for j=1, 2, . . . , r as diagonal elements.

It follows from (A.7) that the eigenvalues λ_(j) (γ)of M are simplyrelated to those of M₀,

    λ.sub.j (γ)=λ.sub.j (0)+γ.       (A.16)

To proceed, one uses the transformation (A.10) in eq. (A.5) to write,

    a.sub.γ =M.sup.-1 d=U D.sub.γ.sup.-1 U.sup.t d, (A.17)

and similarly using eq. (A.15) and (A.6),

    a.sup.(0) =M.sub.0.sup.-1 d.sup.(0) =U.sub.r D.sub.0.sup.-1 U.sub.r.sup.t d.sup.(0).                                                (A.18)

At this point, there are several ways to proceed which all lead to thesame results. The most direct approach is to use the above equations,and write the solutions in eqs. (A.17) and (A.18) as eigenvectorexpansions, i.e., from (A.17) one finds ##EQU61## are the projections ofthe data onto the eigenvectors. Likewise from (A.18), ##EQU62##

In obtaining the above equation, I have used (A.8)and have defined theprojections Q₀,i =u_(i) ^(t) ·n of the noise onto the non-null space ofM₀. The criterion for selecting γ is that the squared norm of thedifference vector, a.sub.γ -a⁰, be a minimum. Therefore, one is led tothe minimization of the function F.sub.γ where

    F.sub.γ =(a.sub.γ -a).sup.t ·(.sub.γ -a.sup.0), (A.22)

where (·) denotes the ordinary scalar product.

Combining eqs. (A.19)-(A.22) and using the orthonormality conditions onefinds that ##EQU63## where C₁ is a constant independent of γ, i.e.,##EQU64##

To proceed with the minimization of F.sub.γ, it is necessary to select adirection for the vector Q₀. The function is maximized (minimized) withrespect to the noise by choosing Q₀ parallel (anti-parallel) to Q. Aconservative approach, also followed by Butler, Reeds and Dawson⁹ is toassume that the vectors are parallel. This assumption is not rigorouslyjustifiable but errs on the side of over smoothing (selecting too largea γ) which is much less dangerous than under smoothing which results ina wildly oscillatory and meaningless inverse. Thus one is lead to write,

    Q.sub.0 =s Q,                                              (A.25),

where the scalar s is determined below.

Requiring that the derivative of F.sub.γ with respect to γ vanish, leadsto a trancendental equation whose solution determines γ_(opt), i.e.,##EQU65##

According to the criterion used in this Appendix, an optimal value of γcan be found by finding the non-negative roots of eq. (A.26). Theequation is solved numerically, by using Newton's method. Note that, byinspection of (A.26) that for noise free data (i.e., s=0) that γ=0 asrequired. From (A.26) one can prove that the equation always has aunique non-negative solution for s<1 which is bounded in the interval,

    γ.sub.l <γ.sub.opt <γ.sub.u,             (A.30)

where γ_(l) =sγ.sub.γ (0) and γ_(u) =sγ₁ (0).

To complete this Appendix, it remains to determine the scalar s in eq.(A.25). To this end, note that ##EQU66##

Using, (A.19a) one finds that, ##EQU67## where the d_(i) are thecomponents of the data vector defined in (A.3) and, ##EQU68##

Similarly, using (A.21) one finds that, ##EQU69## where the n_(i) arethe components of the noise vector defined in (A.9) and where it hasbeen assumed that in computing s one can replace the random variable Q₀^(t) ·Q₀ by its expectation value.

Finally, using (A.9) and the statistical properties of the noise onefinds that, ##EQU70##

APPENDIX B: PROOF THAT M IS POSITIVE DEFINITE

Recall from (A.7) that,

    M.sub.l,k =M.sub.l,k.sup.(0) +γδ.sub.l,k,      (B.1)

where from (67) and (A.4), ##EQU71##

Note that M_(l),k.sup.(0) can be written in the form, ##EQU72## where Ihave defined the matrix, ##EQU73##

For an arbitrary vector VεR^(N).sbsp.3.sup.×1, note that

    V.sup.t B.sup.t BV≡∥BV∥.sup.2 ≧0,(B.5)

so that M.sup.(0) is positive semi-definite. More specifically, since Vis arbitrary, one can choose it to be any eigenvector of M.sup.(0),e.g., u_(j) from which it follows that γ_(j) (0)≧0. For γ>0, it followsfrom (A.15) that γ_(j) (γ)>0 so that M is positive definite.

APPENDIX C: CALCULATION OF Δφ

Recall from eq. (44) that the free-fluid porosity is defined by,##EQU74## where K_(tool) is the tool constant.

This Appendix derives the correction Δφ in the above equation. Thiscorrection is needed because the bound-fluid porosity cut-off T_(c)generally does not lie on the leftmost endpoint of the free-fluidporosity integration interval. The correction is in practive almostnegligble. It does not affect φ_(nmr), only its partitioning into freeand bound fluid porosity. It can be determined exactly by specifying theclosed interval [T_(min), T_(max) ] and the number of integration pointsN_(s) which determines a set of logarithmically spaced T₂,l values forl=1, 2, . . . , N_(s). The l-th integration element is a rectangularstrip of height P(T₂,l), width δ_(l) and area a_(l) ≡P(T₂,l)δ_(l) where,##EQU75##

The integer N_(c) in (C.1) is defined such that T₂,1 ≧T_(c) for l=N_(c),. . . , N_(s). The endpoints of the rectangular strips are at themidpoints of adjacent values of T₂,1. For example, let τ_(l) denote themidpoint (also the leftmost inegration point for the area elementa_(l)). Then by definition, ##EQU76## Note that the widths (δ_(l)) ofthe rectangular strips can be written in terms of the τ₁, i.e.,

    δ.sub.l =τ.sub.l+1 -τ.sub.l.                 (C.4)

This above described picture is shown schematically in FIG. 26 for thetwo rectangular integration elements corresponding to l=N_(c) -1 andl=N_(c).

In general, there are two cases to consider: (a) τ≦T_(c) or (b) τ>T_(c).Case (a) is illustrated in FIG. 26. In case (a), the correction Δφ≦0, sothat the correction subtracts porosity from the summation in (C.1) whichincludes a small amount of bound fluid porosity. The porosity correctionis the sand shaded area in FIG. 26 multiplied by the tool constant. Thiscorrection increases the bound fluid porosity by an equal amount.

A simple calculation shows that for case (a): ##EQU77## and for case (b)one finds that, ##EQU78##

The invention being thus described, it will be obvious that the same maybe varied in many ways. Such variations are not to be regarded as adeparture from the spirit and scope of the invention, and all suchmodifications as would be obvious to one skilled in the art are intendedto be included within the scope of the following claims.

I claim:
 1. A signal processing system adapted for extracting specificformation evaluation information from a set of measured spin echo dataacquired from a logging tool disposed in a borehole, comprising:firstmeans adapted to be disposed within said logging tool in said boreholefor compressing and eliminating redundant information from said set ofmeasured spin echo data and generating a plurality of window sums, wherethe number of said plurality of window sums is less than the number ofsaid set of measured spin echo data, said plurality of window sumsadapted to be transmitted uphole from said logging tool; second meansadapted to be disposed on a surface of said borehole, connected to saidlogging tool and responsive to the plurality of window sums transmitteduphole from said logging tool for generating said specific formationevaluation information; and means for recording said specific formationevaluation information on an output record medium.
 2. The signalprocessing system of claim 1, wherein said first means generates aplurality of signal plus noise amplitudes A_(j).sup.(+) from said set ofmeasured spin echo data, where the number of said plurality of signalplus noise amplitudes is less than the number of said set of measuredspin echo data,said first means generating said plurality of window sums(I_(m),m+1) from said plurality of signal plus noise amplitudesA_(j).sup.(+), where the number of said plurality of window sums is lessthan the number of said plurality of signal plus noise amplitudes,whereby said first means compresses and eliminates redundant informationfrom said set of measured spin echo data when said plurality of windowsums (I_(m),m+1) is generated from said measured spin echo data.
 3. Thesignal processing system of claim 2, wherein said specific formationevaluation information comprises:a set of information includingporosity, porosity standard deviation, free fluid standard deviation,and measurement diagnostics, said set of information being recorded onsaid output record medium.
 4. The signal processing system of claim 2,wherein said specific formation evaluation information comprises:a setof information including porosity, spin spin relaxation timedistributions, and continuous permeability logs, said set of informationbeing recorded on said output record medium.
 5. A nuclear magneticresonance logging system, comprising:a logging tool adapted to bedisposed in a wellbore, said logging tool including, antenna means forsensing induced voltages representative of a plurality of magneticmoments associated with a plurality of protons precessing in a formationtraversed by said wellbore and generating a plurality of spin echoreceiver voltage pulses representative of the magnetic moments,amplitude generating means responsive to said plurality of spin echoreceiver voltage pulses for generating a plurality (J) of inphase(R_(j)) amplitudes and a plurality (J) of quadrature (X_(j)) amplitudes,the plurality of inphase amplitudes and the plurality of quadratureamplitudes totalling 2J amplitudes, and first means responsive to theinphase (R_(j)) and quadrature (X_(j)) amplitudes for generating aplurality (N_(w)) of window sums (I_(m),m+1), the plurality (N_(w)) ofwindow sums being less in number than said 2J amplitudes; processingmeans adapted to be disposed at a surface of said wellbore andresponsive to said plurality of window sums generated by the first meansof said logging tool for generating log outputs and signaldistributions; and output record generating means for recording said logoutputs and said signal distributions representing color maps on anoutput record medium.
 6. The logging system of claim 5, wherein saidfirst means generates an estimate of RMS noise in response to theinphase and quadrature amplitudes,said processing means determining aplurality of standard deviations in accordance with the estimate of RMSnoise, said output record generating means recording said plurality ofstandard deviations on said output record medium.
 7. The logging systemof claim 6, wherein said processing means determines a logarithm of alikelihood function from the estimate of RMS noise and furtherdetermines a set of spectral amplitudes {a₁ } from the logarithm of thelikelihood function,said processing means generating said log outputsand said signal distributions in response to said spectral amplitude. 8.The logging system of claim 5, wherein said first means:estimates asignal phase (theta) in response to the inphase and quadratureamplitudes generated by the amplitude generating means; determines aplurality of signal plus noise amplitudes A_(j).sup.(+) from saidplurality of inphase amplitudes, said plurality of quadratureamplitudes, and said signal phase, said plurality of signal plus noiseamplitudes being comprised of a plurality of groups of signal plus noiseamplitudes; and generates said plurality of window sums from therespective plurality of groups of signal plus noise amplitudes.
 9. Thelogging system of claim 8, wherein said first means determines aplurality of amplitudes A_(j).sup.(-) from said plurality of inphaseamplitudes, said plurality of quadrature amplitudes, and said signalphase,said first means generating an estimate of RMS noise from saidplurality of amplitudes A_(j).sup.(-).
 10. The logging system of claim9, wherein said processing means determines a plurality of standarddeviations in accordance with the estimate of RMS noise,said outputrecord generating means recording said plurality of standard deviationson said output record medium.
 11. The logging system of claim 10,wherein said processing means determines a logarithm of a likelihoodfunction from the estimate of RMS noise and further determines a set ofspectral amplitudes {a₁ } from the logarithm of the likelihoodfunction,said processing means generating said log outputs and saidsignal distributions in response to said spectral amplitude.
 12. Amethod of transmitting data from a well tool disposed in a wellbore to aprocessing system disposed at a surface of said wellbore, said well toolincluding an antenna for sensing information and generating a pluralityof inphase amplitudes and a plurality of quadrature amplitudes inresponse to said information, comprising the steps of:in said well tool,estimating a signal phase from the inphase and quadrature amplitudes;determining a plurality of values A_(j).sup.(+) from the signal phase,the plurality of inphase amplitudes, and the plurality of quadratureamplitudes; sub-dividing said plurality of values into groups therebyproducing a plurality of groups of said values, where the plurality ofgroups is less than the plurality of values; generating a window sumvalue for each group of the plurality of groups of said values, therebyproducing a plurality of window sums, where the plurality of window sumsis equal to the plurality of groups of said values; and transmitting theplurality of window sums uphole from the well tool disposed in saidwellbore to the processing system disposed at the surface of saidwellbore.
 13. In a logging system including a well tool adapted to bedisposed within a wellbore and a processing system disposed at a surfaceof said wellbore, a method of generating an output record mediumillustrating parameters representative of the quality of a reservoirtraversed by said wellbore, comprising the steps of:(a) in said welltool, sensing induced voltages representative of a plurality of magneticmoments associated with a plurality of protons precessing in saidreservoir traversed by said wellbore and generating a plurality of spinecho receiver voltage pulses representative of the magnetic moments; (b)generating a plurality (J) of inphase (R_(j)) amplitudes and a plurality(J) of quadrature (X_(j)) amplitudes in response to said plurality ofspin echo receiver voltage pulses, the plurality of inphase amplitudesand the plurality of quadrature amplitudes totalling 2J amplitudes, and(c) generating a plurality (N_(w)) of window sums (I_(m),m+1) inresponse to the inphase (R_(j)) and quadrature (X_(j)) amplitudes, theplurality (N_(w)) of window sums being less in number than said 2Jamplitudes; (d) in said processing system, generating log outputs andsignal distributions representing color maps in response to saidplurality of window sums; and (e) recording said log outputs and saidsignal distributions representing color maps on an output record medium.14. The method of claim 13, wherein the generating step (c) comprisesthe step of:(f) generating an estimate of RMS noise and said plurality(N_(w)) of window sums (I_(m),m+1) in response to said inphase (R_(j))and quadrature (X_(j)) amplitudes.
 15. The method of claim 14, whereinthe generating step (d) comprises the step of:(g) generating a pluralityof standard deviations in response to the estimate of RMS noise, saidlog outputs and said signal distributions being generated in response tosaid estimate of RMS noise and said plurality of window sums.
 16. Themethod of claim 15, wherein the recording step (e) comprises the stepof:(h) recording said plurality of standard deviations in addition tosaid log outputs and said signal distributions representing color mapson said output record medium.
 17. The method of claim 14, wherein thegenerating step (d) comprises the step of:(i) generating a plurality ofstandard deviations in response to the estimate of RMS noise; (j)determining a logarithm of a likelihood function from the estimate ofRMS noise; (k) further determining a set of spectral amplitudes {a₁ }from the logarithm of the likelihood function; and (l) generating saidlog outputs and said signal distributions in response to said spectralamplitude.
 18. The method of claim 17, wherein the recording step (e)comprises the step of:(m) recording said plurality of standarddeviations in addition to said log outputs and said signal distributionsrepresenting color maps on said output record medium.
 19. The method ofclaim 13, wherein said generating step (c) comprises the stepsof:estimating a signal phase (theta) in response to the inphase andquadrature amplitudes; determining a plurality of signal plus noiseamplitudes A_(j).sup.(+) from said plurality of inphase amplitudes, saidplurality of quadrature amplitudes, and said signal phase, saidplurality of signal plus noise amplitudes being comprised of a pluralityof groups of signal plus noise amplitudes; and generating said pluralityof window sums from the respective plurality of groups of signal plusnoise amplitudes.
 20. The method of claim 19, wherein said generatingstep (c) further comprises the steps of:determining a plurality ofamplitudes A_(j).sup.(-) from said plurality of inphase amplitudes, saidplurality of quadrature amplitudes, and said signal phase, andgenerating an estimate of RMS noise from said plurality of amplitudesA_(j).sup.(-).
 21. The method of claim 20, wherein the generating step(d) comprises the step of:determining a plurality of standard deviationsin accordance with the estimate of RMS noise.
 22. The method of claim21, wherein the generating step (d) further comprises the stepsof:determining a logarithm of a likelihood function from the estimate ofRMS noise; determining a set of spectral amplitudes {a₁ } from thelogarithm of the likelihood function; and generating said log outputsand said signal distributions in response to said spectral amplitude.23. The method of claim 22, wherein the recording step (e) comprises thestep of:recording said plurality of standard deviations in addition tosaid log outputs and said signal distributions representing color mapson said output record medium.
 24. A system for receiving inducedvoltages associated with a material representative of a plurality ofmagnetic moments associated with a plurality of protons precessing insaid material and generating an output record medium for recordinginformation pertaining to a set of characteristics of said material,comprising:means responsive to said induced voltages for generating aplurality of spin echo receiver voltage pulses representative of themagnetic moments; amplitude generating means responsive to saidplurality of spin echo receiver voltage pulses for generating aplurality (J) of inphase (R_(j)) amplitudes and a plurality (J) ofquadrature (X_(j)) amplitudes, the plurality of inphase amplitudes andthe plurality of quadrature amplitudes totalling 2J amplitudes; firstmeans responsive to the inphase (R_(j)) and quadrature (X_(j))amplitudes for generating a plurality (N_(w)) of window sums(I_(m),m+1), the plurality (N_(w)) of window sums being less in numberthan said 2J amplitudes; processing means responsive to said pluralityof window sums generated by the first means for generating log outputsand signal distributions; and output record generating means forrecording said log outputs and said signal distributions on said outputrecord medium.
 25. The system of claim 24, wherein said first meansgenerates an estimate of RMS noise in response to the inphase andquadrature amplitudes,said processing means determining a plurality ofstandard deviations in accordance with the estimate of RMS noise, saidoutput record generating means recording said plurality of standarddeviations on said output record medium.
 26. Th system of claim 25,wherein said processing means determines a logarithm of a likelihoodfunction from the estimate of RMS noise and further determines a set ofspectral amplitudes {a₁ } from the logarithm of the likelihoodfunction,said processing means generating said log outputs and saidsignal distributions in response to said spectral amplitude.
 27. Thesystem of claim 24, wherein said first means:estimates a signal phase(theta) in response to the inphase and quadrature amplitudes generatedby the amplitude generating means; determines a plurality of signal plusnoise amplitudes A_(j).sup.(+) from said plurality of inphaseamplitudes, said plurality of quadrature amplitudes, and said signalphase, said plurality of signal plus noise amplitudes being comprised ofa plurality of groups of signal plus noise amplitudes; and generatessaid plurality of window sums from the respective plurality of groups ofsignal plus noise amplitudes.
 28. The system of claim 27, wherein saidfirst means determines a plurality of amplitudes A_(j).sup.(-) from saidplurality of inphase amplitudes, said plurality of quadratureamplitudes, and said signal phase,said first means generating anestimate of RMS noise from said plurality of amplitudes A_(j).sup.(-).29. The system of claim 28, wherein said processing means determines aplurality of standard deviations in accordance with the estimate of RMSnoise,said output record generating means recording said plurality ofstandard deviations on said output record medium.
 30. The system ofclaim 29, wherein said processing means determines a logarithm of alikelihood function from the estimate of RMS noise and furtherdetermines a set of spectral amplitudes {a₁ } from the logarithm of thelikelihood function,said processing means generating said log outputsand said signal distributions in response to said spectral amplitude.